'%i 
 
THE LONDON SCIENCE CLASS-BOOKS 
 
 ELEMENTARY SERIES 
 EDITED BY 
 
 PROF. G. C. FOSTER, F.R.S. AND PHILIP MAGNUS, B.Sc. B.A. 
 
 I 
 
 I 
 
 ELEMENTARY GEOMETRY 
 
 COXCRCEXT FIG r RES 
 
 '» ^ *-i 
 
 V"-;:,!: 
 
LONDON : I'RINTED BY 
 
 SPOTTISWOODE AND CO., NEW-STREET SQUARE 
 
 AND PARLIAMENT STREET 
 
ELEMENTARY GEOMETRY 
 
 CONGRUENT FIGURES 
 
 BY 
 
 OLAUS HENRICI. Ph.D. F.R.S. 
 
 PROFESSOR OF PURE .MATHEMATICS IN L-NIVEKSITV COLLEGE LONDON 
 
 LONDON 
 
 LONGMANS, GREEN, AND CO. 
 
 1879 
 
 All rights reserved 
 
Sdref^ 
 
EDITORS' PREFACE. 
 
 Notwithstanding the large number of scientific 
 works which have been published within the last few 
 years, it is very generally acknowledged by those who 
 are practically engaged in Education, whether as 
 Teachers or as Examiners, that there is still a want of 
 Books adapted for school purposes upon several im- 
 portant branches of Science. The present Series 
 will aim at supplying this deficiency. The works 
 comprised in the Series will all be composed with 
 special reference to their use in school-teaching ; 
 but, at the same time, particular attention will be 
 given to making the information contained in them 
 trustworthy and accurate, and to presenting it in 
 such a way that it may serve as a basis for more 
 advanced study. 
 
 In conformity with the special object of the 
 Series, the attempt will be made in all cases to bring 
 out the educational value which properly belongs to 
 the study of any branch of Science, by not merely 
 treating of its acquired results, but by explaining as 
 
vi Editors' Preface. 
 
 fully as possible the nature of the methods of inquiry 
 and reasoning by which these results have been 
 obtained. Consequently, although the treatment of 
 each subject will be strictly elementary, the funda- 
 mental facts will be stated and discussed with the 
 fulness needed to place their scientific significance in 
 a clear light, and to show the relation in which they 
 stand to the general conclusions of Science. 
 
 In order to ensure the efficient carrying-out of the 
 general scheme indicated above, the Editors have 
 endeavoured to obtain the co-operation, as Authors 
 of the several treatises, of men who combine special 
 knowledge of the subjects on which they write with 
 practical experience in Teaching. 
 
 The volumes of the Series will be published, if 
 possible, at a uniform price of i^-. dd. It is intended 
 that eventually each of the chief branches of Science 
 shall be represented by one or more volumes. 
 
 G. C. F., 
 P. M. 
 
AUTHOR'S PREFACE. 
 
 Geometry is the science of Space. In its fullest 
 meaning, it embodies the knowledge and the investi- 
 gation of properties of space, however simple and 
 famihar or however intricate they may be, and by 
 whatever process this knowledge may have been 
 acquired. 
 
 To it belongs the knowledge gained by a systema- 
 tic study, which, begun by Egyptian priests in ages 
 long gone by and continued through many centuries, 
 is at the present day carried on more vigorously 
 than ever ; to it also belongs the knowledge uncon- 
 sciously obtained by us while living and moving in 
 space. iVs everything we do or perceive is in space, 
 a large amount of our experience must of necessity 
 relate to it. A carpenter or a mechanician may, dur- 
 ing the pursuit of his calling, obtain a large stock of 
 real geometrical knowledge without knowing anything 
 about a proof, or without even kno^ving that his know- 
 ledge has anything to do with Geometry. 
 
 Experience developes principally the faculty of 
 
viii Author's Preface. 
 
 realising to the mind arrangements of things in space, 
 and the art of deaHng practically with them. 
 
 The ' science ' of Geometry, on the other hand, 
 requires a systematic analysis of the properties of 
 space, beginning with a study of the simplest and 
 going on to that of the most complicated figures, — 
 the simplest figures being not necessarily those with 
 which we meet most frequently in everyday experience. 
 Further, each new property stated has to be shown 
 by a rigid proof actually to contain a general geome- 
 trical truth. 
 
 The two modes of acquiring geometrical knowledge 
 thus indicated, different as they are in their nature, 
 nevertheless react continuously upon each other. 
 
 The ' science ' sprung originally from a desire to 
 systematise previous, and to guide further, experience. 
 The axioms which form its very basis, are obtained 
 by experience, and its study remains barren without 
 constant recourse to the inspection of solids or their 
 substitutes, geometrical figures. Reciprocally, the 
 faculty above referred to of comprehending figures in 
 space and forming clear mental pictures of them, and 
 the art of making practical use of this faculty, are 
 much assisted by the methodical study of Geometry. 
 
 In teaching, both sides ought to be kept in view. 
 The study of the science of Geometry can only be 
 carried on satisfactorily if the student possesses a suf- 
 ficient amount of knowledge gained by experience. 
 
Authors Preface. ix 
 
 Where this is wanting, or where the connection be- 
 tween his experience and the science is not brought 
 home to him, the student will be unable to make any 
 progress : in most cases, I believe, not because he is 
 unable to understand exact reasoning, but simply 
 because he cannot connect the subject reasoned about 
 with any concrete notions he has already acquired. 
 This is, in my opinion, the reason why so many boys 
 fail to understand Euclid, and fall back in desperation 
 on the expedient of learning propositions by heart. 
 This lack of concrete geometrical notions could not 
 exist if all children, either in a Kmdergarte/i or in 
 their play at home, were early made familiar with the 
 simplest forms and their most obvious mutual rela- 
 tions. 
 
 In order, however, to refer constantly to the con- 
 crete, geometrical dramng ought to be combined 
 systematically \vith the teaching of Geometry. This 
 is scarcely possible in connection with Euclid ; and a 
 student who wishes to make practical use of Geometry 
 has to make a separate study of geometrical drawing, 
 whether or not he has already studied and mastered 
 EucHd. Geometrical drawing belongs, in fact, to a 
 branch of Geometry' of which Euclid knew nothing, 
 and where Euclid's propositions are of little use. This 
 branch, itself the outcome of an attempt made nearly 
 a century ago to systematise the art of drawing as 
 developed during past ages by handicraftsmen, is now 
 
X Author'' s Preface. 
 
 known as projective or modern Geometry, and has 
 assumed such dimensions as almost to in chide the 
 whole science. 
 
 I have tried to put the subject in such a manner 
 that the student may thoroughly master the first 
 elements of Geometry, and that he may realise the 
 geometrical contents of the propositions as properties 
 of space through actually seeing their truth by the 
 mental or physical inspection of figures, instead of 
 being convinced of their truth by a long process of 
 logical reasoning. To attain this object it is neces- 
 sary in the proof of each proposition to go back 
 to first principles as far as this is possible ; and this 
 may be done to a very great extent by introducing a 
 notion which is extremely simple, though of great 
 generality. This notion is that of the correspondence 
 of points or lines in two figures which are identically 
 equal or ' congruent,' corresponding points or lines 
 being defined as points or lines which coincide 
 when the figures themselves are made to coincide. 
 This notion is afterwards easily extended to similar 
 figures, and more generally to figures which are pro- 
 jective. 
 
 Just as in higher Geometry the investigation of 
 projective figures is simplified by placing them in per- 
 spective positions, so I bring congruent figures not 
 into a position of coincidence, but into a position of 
 symmetry, which is the perspective position of con- 
 
Atcthors Preface. xi 
 
 gruent figures. This can be done in two ways recip- 
 rocal to each other, and thus the principle of duaUty 
 is ilkistrated at an early period by an important 
 example. This principle I have introduced from 
 the beginning in pointing out the reciprocity between 
 angles and segments of lines, that is, of straight lines 
 of finite length, and I have dwelt on its importance 
 throughout the book. For this reason I have intro- 
 duced the figures reciprocal to loci by considering 
 what I call ' sets of lines,' i.e. the aggregate of all 
 lines which satisfy a given condition, and I have con- 
 sidered the circle first as a locus of points and then 
 as the envelope of a set of lines. 
 
 Another innovation consists in the early intro- 
 duction of the notion of ' sense ' in a line or an 
 angle, corresponding to the signs (+ and — ) in 
 algebra. 
 
 A number of exercises has been added to each 
 chapter. Many of these refer to geometrical drawing, 
 and include instructions about the use of instruments. 
 Besides the straight-edge and a pair of compasses, 
 the student is supposed to possess the two usual 
 kinds of set squares. Preference is given to those 
 constructions which are performed by aid of straight- 
 edge and set squares, the use of the compasses being 
 avoided as much as possible. 
 
 Some of the exercises refer to the logic of the 
 proofs, and especially to the logical connection be- 
 
xii Author's Preface. 
 
 tween propositions with reference to a ' Digression 
 on Logic ' which has been inserted between Chapters 
 11. and ILL 
 
 To give a prehminary notion of the order in 
 which the subject has been arranged, I add a short 
 statement of the contents of the several chapters. 
 
 The first chapter contains the fundamental notions 
 of Geometry, viz. those of point, curve, surface, solid, 
 and space, and the first three axioms. Next the 
 straight Hne and the plane are considered, and two 
 more axioms are given. On this follow figures, con- 
 sisting of either two points or two lines which inter- 
 sect. The two points give rise to a finite straight 
 line called a segment, the two lines to an angle. A 
 separate chapter is devoted to the bisections of such 
 segments and angles. Special attention has been 
 paid to fixing the ' sense ' of a segment or an angle, 
 and some of the simplest of Mobius' equations have 
 been stated. Next, parallel lines are investigated, and 
 the sixth and last axiom is introduced. The theorems 
 about angles in polygons, which are immediate con- 
 sequences of the theory of parallels, follow, and give 
 first theorems about figures consisting of more lines 
 and points. With this the foundation of Geometry 
 has been laid. The fundamental notions, the axioms, 
 and the simplest figures are given, and the study of 
 more complicated figures can be undertaken. 
 
 The rest of the volume is devoted to the investi- 
 
Author's Preface. xlli 
 
 gation of the theory of figures which are equal m all 
 respects, and which I call congruent, and to conse- 
 quences following from this theory. It coincides 
 in its contents with the first four books of 
 Euclid, with the exception of the theorems relating 
 to areas. The treatment, however, is verj^ different. 
 In the eighth chapter congruent figures, which are 
 defined as figures that can be made, to coincide, 
 are considered quite generally. It is pointed out 
 that to every point or line in one 'corresponds' a 
 point or a line in the other, and generally that to 
 every part in the one there is a ' corresponding ' part 
 in the other. Then congruent figures are brought 
 into special positions. They are first made to coin- 
 cide, and then in their common plane either a fine or 
 a point is fixed. The one figure is now with its 
 plane turned about the fixed line till it falls again 
 into the plane, or it is turned in its plane about the 
 fixed point. The figures are then said to be sym- 
 metrical either with regard to an axis or with regard 
 to a centre of symmetr}-. Symmetrical figures are 
 thus congruent figures in special positions ; they 
 are the most special cases of projective figures in 
 perspective position. Some of the most important 
 properties due to this position are next stated. The 
 proofs are so simple that they are in most cases only 
 indicated, or even altogether omitted. Every reader, 
 especially every teacher, will easily complete them if 
 
xiv Author's Preface. 
 
 he has grasped the meaning of correspondence in 
 symmetry. 
 
 After this general investigation, which each teacher 
 can give to his pupils all at once, or only gradually as 
 wanted, the study of special figures is begun. First 
 single triangles and quadrilaterals are considered, the 
 beginning being made with the symmetrical triangle, 
 which of course is Euclid's isosceles triangle. From 
 the fact that it has an axis of symmetry all its 
 properties follow at once. The theorem that the 
 greater side in a triangle is opposite the greater 
 angle, and others connected with it, follow with equal 
 ease. 
 
 Of symmetrical quadrilaterals there are three, one 
 with a centre, the parallelogram, and two with an 
 axis. Of these, one has a diagonal as axis of sym- 
 metry, and this I call a kite, adopting the name given 
 to it by Prof Sylvester ; the other is the symmetrical 
 or isosceles trapezium. Their properties all follow 
 from their symmetry. 
 
 After this I have inserted a chapter on congruent 
 triangles. Of their theory, however, little use is 
 made in the sequel. Then follows a chapter on 
 * loci of points ' and * sets of lines,' as explained 
 before. The rest of the book is devoted to the circle, 
 which is first treated as a locus and then as the enve- 
 lope of a set of lines. The circle being the embodi- 
 ment of symmetry, I need not dwell on the ease with 
 
AuiJioj's Preface. xv 
 
 which its properties follow from considerations based 
 on symmetry. 
 
 I have undertaken this book from the desire to 
 prepare students from the verj' first for those modern 
 methods of which the method of projection and the 
 principle of duality are the most fundamental. The 
 advantages of the method adopted ^\all, however, be 
 fully appreciated only in their continuation in tlie 
 second volume, which will treat of areas in con- 
 nection with what Mobius calls ' equal figures ' and 
 of similar figures. These figures in their persj^ective 
 positions follow from the two kinds of symmetrical 
 figures, by dropping one measurement in each 
 case. From axial symmetry we obtain ' equal ' or 
 skew-symmetrical figures by drawing the lines join- 
 ing corresponding points not perpendicular to the 
 axis, whilst similar figures similarly situated are ob- 
 tained from central symmetry by dropping the con- 
 dition that corresponding points shall be equidistant 
 from the centre. 
 
 O. Hexrici. 
 
 University Collkgi:, London : 
 December, 1878. 
 
I 
 
 CONTENTS. 
 
 CHAPTER I. 
 
 FUNDAMENTAL NOTIONS. 
 
 PAGE 
 
 Properties considered in Geometry are shape, size, position, 
 and movability of solids — Space— Solids, surfaces, 
 curves, points— Curve described by motion of a point — 
 Sense of motion — Arc or segment of curve, its sense — 
 A surface described by a mo\-ing curve — Spreads and 
 dimensions — Continuity — Solid described by moving 
 surface — Axiom I. Space is of three dimensions — Defi- 
 nition of figure — Coincident figures — Axiom II. Figures 
 which can be made to coincide here may coincide any- 
 where — Congruent figures — Axiom III. A figure may 
 be moved with one or two points fixed . . . . i 
 
 CHAPTER II. 
 
 LINES AND PLANES. 
 
 Notion of line — A line is of indefinite extension — Axiom 
 IV. Two lines coincide if they have two points in com- 
 mon—Join of points, join of lines— Pencil of lines — 
 Rotation about a line — The plane; is of indefinite ex- 
 tension — Axiom V. Two planes coincide if they have 
 three points in common which do not lie in a line — 
 Intersection of planes, their join — Lines in a plane — 
 Axial pencil (of planes) — Join of plane and line— Three 
 planes — Definition of Plane Geometry . . . -17 
 
 Exercises ......... 27 
 
xviii Contents. 
 
 DIGRESSION ON LOGIC. 
 
 PAGE 
 
 Proposition — Division of things into classes — ^Negative 
 classes — Notation — Contra-positive form of proposition — 
 Converse and obverse proposition — Examples — Geo- 
 metrical propositions : -definitions, axioms, theorems, 
 corollaries, and problems — Axioms define space — Exer- 
 cise .......... 29 
 
 CHAPTER III. 
 
 SEGMENTS AND ANGLES. 
 
 Reciprocal figui-es and theorems — Notation used — Row of 
 points and pencil of lines. Rays and half-rays — Segment 
 of line — Sense of segments, AB= —BA — Sum and dif- 
 ference of segments — Analogy with algebraical addition 
 and subtraction; BC-\- CA +AB = o — Definition of 
 angle; its sense — Angles of continuation and of rota- 
 tion—Equal angles — Adjacent angles ; sum of angles ; 
 al) + ^>c +r(7 = — Negative angles — Vertically opposite 
 angles .......... 36 
 
 Exercises 53 
 
 CHAPTER IV. 
 
 BISECTORS OF SEGMENTS AND ANGLES. 
 
 Bisector or mid-point of a segment — Bisector or mid-ray of 
 an angle — Every segment or angle has one and only one 
 bisector —Analogies and differences between segments 
 and angles — Bisector of sum of two angles or segments 
 — Right angles and perpendiculars — Complementary 
 and supplementary angles. Notation — Adjacent supple- 
 mentary angles — Vertically opposite angles are equal — 
 Bisectors of vertically opi^osite angles . . . • 55 
 
 Exercises ......... 63 
 
[ 
 
 Contents. xix 
 
 CHAPTER V. 
 
 PARALLEL LINES. 
 
 PAGE 
 
 Analysis — Necessity of a new axiom — Definition of parallel 
 lines — Axiom VI, Through a point only one line can be 
 drawn parallel to a given line — Angles formed by two 
 lines cut by a transversal — Properties of these angles — 
 Theorems about parallel lines — Pencil of parallels . . 65 
 
 Exercises . . . , . . . . .74 
 
 CHAPTER VI. 
 
 ANGLES IN POLYGONS. 
 
 Broken line. Sides, sense, vertices, angles — Convex 
 broken line — Exterior angles — Polygon. Sides, vertices, 
 sense, angles. Convex polygon — Names of polygons — 
 Diagonals — Triangle. Sum of angles equal to angle of 
 continuation — Exterior angle equals sum of interior and 
 opposite angles — Sum of angles in convex polygon — Sum 
 of exterior angles of convex polygon . . . .76 
 
 Exercises ......... 84 
 
 CHAPTER VII. 
 
 AXIAL AND CENTRAL SYMMETRY. 
 
 Congruent figures— Correspondence of points, lines, »S:c. — 
 Axial symmetry; Example — Central symmetry; Examples 
 — Reciprocity — Any two congruent figures may be 
 placed in a position of symmetry' —Symmetrical figures. 
 Axes and centres of figures — Properties following from 
 congruence — Properties following from position of sym- 
 metry, relating to axis and centre — Examples of sym- 
 metrical figures. Perpendicular bisector . . .85 
 
 Exercises ......... too 
 
XX Contents. 
 
 CHAPTER VIII. 
 
 THE TRIANGLE. 
 
 PAGE 
 
 Triangle. Notation. Median line — Symmetrical triangle, 
 has a median line as axis — The equilateral triangle — 
 The unsymmetrical triangle — Sum of two sides greater 
 than the third — Inequalities in axial symmetry — Per- 
 pendiculars and obliques — Distance — Number of obHques 
 of given length ........ 104 
 
 Exercises 
 
 CHAPTER IX. 
 
 SYMMETRICAL QUADRILATERALS. 
 
 Quadrilaterals. Diagonals. Median lines — Possible cases 
 of quadrilaterals with an axis of symmetry — The kite. 
 Properties — Conditions that a quadrilateral may be a 
 kite — The symmetrical trapezium. Properties— Condi- 
 tions that a quadrilateral may be a symmetrical trapezium 
 — The parallelogram. Properties — Conditions that a 
 quadrilateral may be a parallelogram — The rhombus — 
 The rectangle— The square 113 
 
 Exercises . . . . . . . . .124 
 
 CHAPTER X. 
 
 CONGRUENCE OF TRIANGLES. 
 
 Positions of symmetry of congruent triangles — The differ- 
 ent cases of congruent triangles — Remarks — Triangles 
 having two sides equal and included angles unequal . 127 
 
 Exercises 133 
 
Contents. xxi 
 
 CHAPTER XI. 
 
 LOCI OF POINTS AND SETS OF LINES. 
 
 PAGE 
 
 Two conditions required to determine a point or line — De- 
 finitions of loci of points and of sets of lines, their recip- 
 rocity — Examples — Locus of points equidistant from 
 two points; set of lines equally inclined to two lines- 
 Points equidistant from two lines. Lines equidistant 
 from two points— Use of loci — Points equidistant from 
 three points — Points equidistant from three lines. Lines 
 equidistant from three points — Properties of triangles — 
 Number of lines in a point equidistant from another 
 point 135 
 
 Exercises ......... 150 
 
 CHAPTER XIL 
 
 THE CIRCLE AS A LOCUS. 
 
 Definition of circle ; centre ; radius — Congruent circles — 
 A circle may slide along itself — Intersection of a line and 
 a circle — Secant. Chord — Centre of circle is centre of 
 symmetry — Every diameter is axis of symmetry — Arcs 
 of circle. Circumference — Sense of arc — AB + BC+ CA 
 = 0, (Sic, — Angles at centre-^Equal arcs subtend equal 
 angles at centre. Converse — Semicircle. Quadrant — 
 Diameter bisects chords perpendicular to it ; bisects arcs 
 and angles at centre — Tangents — Line perpendicular to 
 radius through end point is a tangent — Tangent as 
 limit of chord — Angle at the circumference defined — 
 Supplementary arcs defined — Angles at circumference 
 upon the same arc or on equal arcs are equal — Two 
 angles at circumference standing on supplementary arcs 
 are supplementary, &c.— Other theorems . . .151 
 
 Exercises ...,,.. .167 
 
XX ii Conients. 
 
 CHAPTER XIII. 
 
 THE CIRCLE AS ENVELOPE. 
 
 PAGE 
 
 Envelope of set of lines equidistant from a point is a circle 
 — Reciprocity between locus and envelope — Tangents 
 from a point without a circle — Construction of tangents 
 from a point without a circle . . . . .169 
 
 Exercises 174 
 
 CHAPTER XIV. 
 
 CONDITIONS DETERMINING A CIRCLE. 
 
 Circles through two points or touching two lines — Circle 
 through three points. Three points determine a circle. 
 Three lines determine four circles — Two circles have 
 not more than two points in common— Concentric 
 circles — Axis of symmetry of two circles — Common 
 chord — Contact of circles. Point of contact, common 
 tangent — Possible positions of two circles — Two circles 
 may have four common tangents— Four concyclic points 
 — Quadrilaterals inscribed in a circle — Quadrilaterals 
 circumscribed about a circle 176 
 
 Exercises ......... 186 
 
ELEMENTS 
 
 OF 
 
 PLANE GEOMETRY. 
 
 PART I. 
 
 CONGRUENT FIGURES. 
 
 CHAPTER I. 
 
 FUNDAMENTAL NOTIONS. 
 
 § I. If we set ourselves the problem of investigat- 
 ing the properties of things which we observe, we are 
 very soon led to subdivide it into a number of dif- 
 ferent problems, according to the nature of the pro- 
 perties on which we fix our attention. This leads to 
 the subdivision of natural science into various branches. 
 Of these the science of mathematics is the simplest, 
 and, consequently, it has reached the highest stage 
 of development. It treats of the most general pro- 
 perties of things. The first notion which anything 
 suggests to us is that of its own existence. If in con- 
 sidering anything we take account only of the fact of 
 its existence, we obtain the notion of a unit ; and by 
 considering the existence of things in a group we get 
 u 
 
2 Elements of Plane Geometry. 
 
 the notion of a group of units — that is, of a munber. 
 This leads to the sciences of Arithmetic and Algebra, 
 with which, however, we are not concerned at present. 
 They alone constitute, strictly speaking, pure mathe- 
 matics. As we gradually take more and more pro- 
 perties into consideration we are led in succession to 
 branches of science of greater and greater complexity. 
 
 Thus in Geometry we consider the shape, size, posi- 
 tion, and motion of things. The introduction of the 
 notion of Time leads us from Geometry to Kinematics. 
 From this science we are brought to that of Kinetics, 
 or Dynamics, by adding the notions of Matter and 
 Force. Here matter is considered only as having 
 inertia and as being acted upon by force. Other pro- 
 perties of matter which manifest themselves in the 
 phenomena called heat, light, electricity, and so on, 
 are investigated in Physics, whilst in Chemistry the 
 differences in kind of matter are studied. The much 
 more complex conditions and changes of life as ob- 
 served in plants and animals constitute the subject- 
 matter of Biology. 
 
 § 2. Geometry. — Geometry, which is the branch of 
 science with which we are at present concerned, treats 
 of some of the properties which are common to all such 
 things as are cognisable by the senses of touch and 
 sight. 
 
 § 3. Shape.^ — By these senses we are led to per- 
 ceive that bodies differ widely in colour, weight, tem- 
 perature, and in many other properties, all of which 
 depend more or less on the material out of which the 
 bodies are formed. Other properties, however, are in- 
 dependent of the material — as, for example, the shape 
 
Shape, Si£:c, and Position. 3 
 
 of a body. Thus two spheres or globes have the 
 same shape : though the one may be made of iron, 
 the other of wood or marble, still they are globes — 
 that is, things having a peculiar, definite shape. 
 
 § 4. Size. — Again, if we take two solids of the same 
 shape — say, two globes — these need not be equal. 
 The one may be small, the other large. We express 
 this by saying that the two bodies, though of the same 
 shape, are of different sizes. On the other hand, two 
 globes, though of the same size, may be of different 
 material, so that size, like shape, does not depend upon 
 material. 
 
 § 5. Position. — But even if we have two solids of 
 the same size, of the same shape, and similar in all 
 other respects, so that, considering them each by 
 itself, it is impossible to tell which is which, they are 
 still not the same. They are distinguished from each 
 other by occupying dii^^xtxii positions in space. 
 
 Two material bodies cannot occupy the same space. 
 We are thus led to recognise a third property com- 
 mon to all bodies : every body has position. 
 
 § 6. Motion. — This brings us to the last property 
 which we have to consider in geometry. A body may 
 change its position, and may be moved about in space — 
 that is, it may assume different positions at different 
 times. 
 
 § 7. We have thus obtained, by appealing to uni- 
 versal experience, four distinct properties which are 
 common to all bodies, but are independent of their 
 material. These are shape, size, position, and capability 
 of being moved ; and they are the only properties with 
 which geometry is concerned. 
 
4 Elements of Plane Geometry. 
 
 Whatever other properties a body may possess, we 
 leave them out of consideration, and treat them 
 practically as though they did not exist. 
 
 § 8. Space. — The four geometrical properties men- 
 tioned above all refer to space. Space itself is a fun- 
 damental conception which it is impossible to define 
 or even to describe. Everything that we observe is 
 in space, and space extends around us in all directions. 
 
 § 9. We may now define geometry as the science 
 which treats of the properties of space. 
 
 Of space itself we derive the first and fundamental 
 properties from experience. These fundamental pro- 
 perties are laid down in propositions, called axioms. 
 Before these can be stated we must develope the 
 notions of the above-mentioned properties of bodies 
 somewhat more fully. 
 
 § 10. Solids. — We have seen that the geometrical 
 properties of a body do not depend on the matter 
 composing it, and therefore we must consider them as 
 remaining unchanged, even if we could conceive that 
 the matter ceased to exist. 
 
 The embodiment of these residual properties con- 
 stitutes the geometrical notion of a solid. 
 
 When, therefore, the word solid is used in geometry, 
 it is to be taken in this sense. It will be seen that 
 such a geometrical solid can exist only as a mental 
 conception ; it can have no nuaterial existence, but 
 the idea of it is obtained by intellectually abstracting 
 the non-geometrical properties from material bodies. 
 
 A solid may then be defined, or rather described, 
 as a pa7't of space boiifided on all sides. 
 
 § II. Surfaces. — That which bounds a solid and 
 
Surfaces, Orrves, and Points. 5 
 
 separates it from other parts of space is called its sur- 
 face. If, for example, we consider a tumbler with 
 water, then the water, if at rest, will occupy a definite 
 part of space and have a definite shape. The boun- 
 dary of this part of space separates the water at the 
 bottom and the sides from the glass of the tumbler, 
 and on the top from the air in the room. The boun- 
 dary between the water and glass forms part neither 
 of the water nor of the glass, but separates the one 
 from the other, so that where the one ends the other 
 begins. This is expressed by saying a surface has no 
 tJiickuess. 
 
 A surface is in space, but is not a part of it in the 
 same sense as a solid is. It has, however, size, shape, 
 and position, and may be moved. 
 
 § 12. Curves. — A surface may consist of different 
 parts. Thus, one part of the surface of the water in 
 the tumbler consists of the boundary between water 
 and glass, while another part separates the water from 
 the air. Each of these parts is bounded where it 
 meets the other part. 
 
 The boundary of a surface or of part of a surface 
 is called a line or a curve. 
 
 The edges of sohds — the edges of a square box, 
 for instance — are lines. Curves and lines, as well as 
 surfaces, have shape, size, and position, and may be 
 moved with the surface or solid on which they lie. 
 
 § 13. Points. — A line or curve may itself be 
 bounded. 
 
 The edges of a square box meet, and are bounded, 
 at the corners of the box. 
 
 The boundaries of a line or curve are called /^^w//j-. 
 
6 Elements of Plane Geometry. 
 
 A point has neither size nor shape ; but it has posi- 
 tion, and may be moved ; for we may move the 
 solid on which it lies. 
 
 A point, then, as long as we do not move it, has 
 only one property : it has position only, and marks 
 a place in space. 
 
 § 14. We have thus obtained the fundamental 
 notions with which we have to deal in geome- 
 try, viz. the notions of space., of solids, surfaces, curves, 
 and poijits. 
 
 The following considerations will greatly assist in 
 getting a clearer conception of them. 
 
 § 15. As it is the solid of which the notion is given 
 directly by experience, we make it again our starting- 
 point. We suppose that a solid gets reduced in 
 size, and becomes smaller and smaller without limit, 
 till at last it loses all size and, with its size, its shape. 
 This gives us the notion of a point. The only pro- 
 perties which it retains are position and capability oj 
 being moved. Such a point does not exist in the 
 material world. It is, in fact, an abstraction. 
 
 A point has no extension, and this must be care- 
 fully remembered. We may conceive points every- 
 where in space ; their number is unlimited. But if 
 we bring different points together into the same 
 position they will never give us anything but a point; 
 we never obtain any extension. We cannot, therefore, 
 say that space is made up of points, although space 
 contains an unlimited number of them. 
 
 § 16. Path of Moving Point. — But a point may 
 be moved, and then it will describe a path. This 
 path of a 7noTino; point is a curiae. Again, a curve 
 
Sense of a Curve. 7 
 
 contains an unlimited number of points. We may 
 take any number of points on a curve, but, however 
 near we take them, there will be room on the curve 
 for other points between them. When two points 
 come together they coincide and form one point only. 
 
 The notion of a line may be obtained directly 
 by considering a wire bent into any shape and ab- 
 stracting all thickness from it. 
 
 § 17. Sense of Motion. — Suppose now we take 
 two points on a curve — we may distinguish them by 
 calling^ one the point A and the other the point B — 
 then the moving point by which we suppose the curve 
 to be described may either come first ^1^. i. 
 
 to A and afterwards to B, or may first 
 come to B and then to A. The curve 
 may, therefore, be described by the 
 point in two different ways. This is expressed by 
 saying that the point moves along the curve either in 
 one or in the opposite sense. Also the curve itself is 
 said to have a sense, or to be of one or the other sense, 
 according as it is considered as being described by a 
 point moving in the one or in the other sense. For 
 instance, a person may go from London to Brighton, 
 or by the same road from Brighton to London. 
 Going from north to south he would traverse the dis- 
 tance in one sense : going from south to north he 
 would traverse it in the opposite sense. Similarly, 
 the sense of a motion from left to right is opposite to 
 that of a motion from right to left. This difference 
 
 ' A point will in future be denoted by a capital letter, like A, 
 B, C, ox M, F, &c., and, as a rule, capital letters will always 
 denote points, and points only. 
 
8 Elements of Plajic Geometry. 
 
 in sense is often distinguished by calling the one 
 sense positive and the other negative. We may call 
 either of them positive, and then the other is fixed as 
 negative. We shall, as a rule, consider the motion 
 from left to right as positive. 
 
 The sense is generally denoted by an arrow-head. 
 
 § t8. If we consider a point A on sl curve, we may 
 
 move another point from it along the curve either in 
 
 the one sense to B, or in the 
 
 opposite sense to C. These two 
 
 points are said to be on opposite 
 
 sides of A. A lies between C 
 
 and B, and sepai-ates them. A point i?i a curve has, 
 
 therefore, tivo sides. 
 
 § 19. Dimension of a Curve. — The different points 
 in a curve follow each other in such a manner that we 
 can pass in only two different ways from a point A 
 to other points ; we must move either in the one or 
 in the opposite sense. This fact is expressed by 
 saying that a curve is of one dimensio7i. It has length. 
 § 20. Space is Unlimited. — A point, in describing 
 a curve, may either return to its original position — 
 and then the curve is said to be closed^ox the point 
 may move to a greater and greater distance from its 
 first position ; and to this process we cannot, from 
 experience, conceive any limit. Curves, then, may be 
 closed, or they may extend to an indefinite distance. 
 
 This shows also that space itself must be considered 
 to be of an indefinite extension. We cannot, indeed, 
 conceive a limit to space, but neither can we compre- 
 hend anything infinite or of indefinite extension. Our 
 experience is limited and leaves us liere in the dark. 
 
S/ir/dcrs. 9 
 
 § 21. Arc of a Curve. — x\ny part of a curve 
 bounded by two points, A and B, may be called an 
 (7/r, or a segtnent of the curve, and may be denoted 
 by AB or by one small letter, say, a} 
 
 § 2 2. An arc of a curve may be moved without 
 changing its shape or size. It will thus describe a 
 path; this path may in a few special cases be the 
 original curv^e again. This, however, is possible oaly 
 if the moving curve be a part of a straight line, of a 
 circle, or of a helix (the thread of a screw is a helix), 
 for such a cur^'e may slide along itself. 
 
 § 23. Path of a Moving Curve. — In all other 
 cases the path of a moving curve is not a curve, but 
 something difterent which is called a surface. A 
 sinface is the path of a moving ciiii'e. 
 
 § 24. If we consider any two positions succes- 
 sively occupied by the describing curve, and denote 
 these positions by a and b, then the moving curve 
 may either come first to a and afterwards to b^ or it 
 may move in the opposite sense by arriving first at b 
 and then at a. Hence if we take any position, a (fig. 
 3), we may pass to another posi- Pj^ 2. 
 
 tion of our curve by moving from 
 a either in one sense to a 
 position b, or by moving in the 
 opposite sense to a position c. 
 This shows that a curve 07i a 
 surface has tiuo sides, and further that curves generating 
 a surface follow each other like the points on a curve. 
 
 ' A curve will in future be denoted by a small letter — a, b, c, 
 ox p, q, r, &c. — and small letters will, as a rule, be used to denote 
 curves or parts of curves only. 
 
lo ElciJieiits of Plane Geometry. 
 
 § 25. Dimensions of a Surface. — From one posi- 
 tion of the moving curve we may pass to other posi- 
 tions by moving either in one or in the opposite sense. 
 Hence, if we consider the surface as generated by the 
 motion of a curve, we may say the surface is of one 
 dimension with regard to the describing curve, which 
 in this case is considered as an element. The curve 
 is here, in fact, considered not as generated by the 
 motion of a point, but given as a whole. But if we 
 consider the curve as containing points, or as being 
 described by a moving point, it is itself of one dimen- 
 sion, and then the surface is said to be of two dimen- 
 sions with regard to the points as elements. 
 
 § 26. Spreads. — The word 'dimension' has here 
 been used in a meaning different from that generally 
 attached to it. In order to avoid confusion, which 
 might arise by stating of a surface at one time that 
 it has one, at another time that it has two dimensions, 
 a different nomenclature has been proposed. 
 
 We have seen that a curve contains an infinite 
 number of points, winch may be considered as being 
 spread along the curve. Similarly points and curves 
 are spread over the surface, and also throughout 
 space. A curve, or a surface, may therefore be called 
 a spread. A surface is a spread with regard to points, 
 as well as with regard to curves as elements. 
 
 There are thus different kinds oi spreads. A curve, 
 or any other spread on which the elements follow like 
 the points in a line, is said to be one-wayed, or to be 
 a one-way spread. 
 
 Hence a curve is a one-7c>ay spread, with points as 
 elements. A surface, as generated by a moving curve. 
 
Examples of Spreads. 1 1 
 
 is a onc--ivay spread^ with the eurves as eleuiciits ; but 
 a surface is a two-way spread, ivitJi points as eie- 
 ments. 
 
 §27. Dimension. — The word 'dimension' is re- 
 tained to denote the number of ways of a spread, con- 
 sidering the points as elements, and then we may say 
 a euTT'e is of one dimension, called length ; and rt; su?face 
 is of two diniensiojis, called length and breadth. 
 
 § 28. The use of the word ' spread ' will be under- 
 stood more fully if we apply it to non-geometrical 
 notions. 
 
 Time extends from an indefinite past to an in- 
 definite future ; it contains an unlimited number of 
 moments. From any given moment of time we may, 
 mentally, go in two different ways to other moments, 
 either to an earUer or a later moment. The moments 
 of time, therefore, follow each other like points in a 
 curve ; and thus we may say that time is a spread of 
 momefits. As we can pass from one moment to another 
 moment only in one or in the opposite way, it is a 
 one-way spread. Hence time is a one-way spread, 
 with moments as elements. 
 
 In a similar manner the temperature of a body 
 may be changed. This, again, can be done only in 
 one or in the opposite way ; we may either raise or 
 lower it. The change of temperature takes place, 
 therefore, either in one sense or in the opposite sense. 
 
 Temperature, consequently, is a one-way spread, 
 with degrees of temperature as elements. 
 
 In a similar manner the weight of a body is a one- 
 7uay spread, and so on. 
 
 As another interesting example lei us consider a 
 
12 Eltincnts of Plane Geometry. 
 
 musical tone of a certain pitch. AVe may, without 
 changing its pitch, vary it in loudness or intensity ; 
 and, as this may be done in one or in the opposite 
 sense, we have again a one-way spread of variations 
 of intensity. 
 
 But we may also change its pitch in one or in 
 the opposite sense, making the pitch higher or lower. 
 Hence we may change a musical tone in two ways, 
 each giving rise to a spread, which is vtry different 
 from the other. The one is the way along which 
 intensity changes, the other is the way along which 
 pitch changes. We therefore say that a musical tone 
 allows of variations which form a two-way spread, 
 with different degrees of intensity and of pitch as 
 elements. 
 
 The number of examples might be multiplied by 
 considering light and colour ; but the above will be 
 sufficient to illustrate the meaning of the word 'spread.' 
 Only one point has still to be considered. 
 
 § 29. Continuity. — In all the above examples the 
 variation of the elements was of a peculiar kind. A 
 point, in changing its position on a curve, passes, in 
 moving from one position to another, through all 
 intermediate positions. It does not move by jumps. 
 This is expressed by saying that the point changes its 
 position continuously., that the curve is continuous. 
 
 If we take, instead of the curve^ a row of solids — 
 say, peas, laid in a row — we have not a continuous 
 change in going from one to the other. 
 
 In order that an aggregate of elements, whatever 
 these may be, may be called a spread, it is necessary 
 that these elements follow continuously. 
 
Solids. 1 3 
 
 ■ Thus the natural numbers i, 2, 3, 4 . . .do not 
 fomi a spread, as there is not a continuous change 
 between them. 
 
 Aristotle indicated this difference by saying that a row of 
 elements is continuous if the boundary of one element is also 
 the boundary of the next, whilst the row is disjnnctive if two 
 consecutive elements have separate boundaries. 
 
 § 30. A surface, like a curve, may be closed^ if, for 
 instance, the generating curve is closed and returns 
 into its original position ; or it may extend indefinitely^ 
 if we suppose the generating curve to move to an in- 
 definite distance from its first position. 
 
 § 31. If we now consider a surface of finite exten- 
 sion, and we move it about in space, it will either 
 slide along itself or generate something new. 
 
 If a smooth piece of paper be placed on the flat 
 surface of a table, it may be made to slide about that 
 surface, and will thus have a surface as its path. 
 Similarly the surface of a globe or of a cylinder may 
 slide along itself. 
 
 But this is a very special motion, possible only 
 with some special surfaces. In general the moving 
 surface describes a solid. 
 
 § 32. Path of a Moving Surface. — The path of 
 a moving surface is in general a solid. 
 
 \ '^'^. If we fix on one position of the moving 
 surface, the surface may move, in one sense or in the 
 opposite sense, to other positions, as in the former 
 cases. A surface in a solid., therefore., has tivo sides. 
 
 § 34. Dimensions of Space.— Further, a solid is 
 a one-way spread of the moving surface ; but, as the 
 surface is a two-way spread of points, the solid is a 
 
14 Elements of Plane Geometry. 
 
 three-way spread of points. In other words, a solid is 
 of three dimensions. 
 
 If we next move a solid about in space, it will be 
 found that its path will be a solid and that we obtain 
 nothing new. Space itself is a three-way spread of 
 points, or space is of three dimensions. 
 
 That a solid has three dimensions is commonly 
 indicated by saying that a solid has lengthy breadth, 
 and thickness ; but this applies strictly to some solids 
 only. A globe can scarcely be called long or thick, 
 nor can we very suitably speak of the breadth of a 
 wire. The expressions are fully justified only for solids 
 of the shape of a brick and similar shapes. 
 
 § 35. The above considerations have made us 
 acquainted with properties of space. These have 
 been obtained by observation. Collecting them into 
 one proposition, we obtain our first axiom — that is to 
 say, a statement obtained by experience. 
 
 Axiom I. Space is of three dimensions. 
 
 Or, in the language of § 27 : 
 
 Space is a three-7vay spread, icith points as elements. 
 
 This includes the statement that space is con- 
 tinuous (§ 29). 
 
 § 36. Figures. — Solids, surfaces, lines or curves, 
 and points, or any combination of them, will be called 
 geometrical figures, or figures simply. 
 
 A figure has therefore in general all the properties 
 enumerated in § 7. It has shape, size, and position, 
 and it may be moved — that is, it may change its po- 
 sition without changing its shape or size. Shape and 
 size, too, are in some investigations considered as 
 liable to change, but in elementary geometry they are 
 
Axioj/is. 1 5 
 
 generally treated as invariable. This, however, does 
 not prevent us from speaking occasionally of moving 
 points or lines in a figure, although the shape of a 
 figure is thereby changed. 
 
 § 37. Coincidence of Two Pigures. — Two figures 
 which have the same shape, the same size, and the 
 same position are said to coincide or to be coincident. 
 Two coincident figures cannot be distinguished from 
 one another till they are separated by moving one into 
 a different position. If both figures are now moved, 
 each on a separate path, to any other part of space, 
 we may try to make them again coincident. That 
 we shall succeed in this could not be proved before- 
 hand. But experience teaches that if we make one 
 solid fit here and now into the hollow of another, it 
 will also fit at any other time and at any other part of 
 space. Thus the different parts of a machine which fit 
 here, will fit if they are sent to India, and this equally 
 whether they are sent all by the Suez Canal, or whether 
 some are sent by this route and others round the Cape. 
 If they do not fit, we say that one or other of the parts 
 has changed its shape or size ; and we account for it 
 by changes in the material produced by bending, by 
 change of temperature, or by some other physical 
 cause. The only criterion which we have as to the 
 invariability of shape and size consists in the fitting in 
 of material solids into hollows made in other solids. 
 
 § 38. This important result ot our experience we 
 shall consider as applicable to geometrical figures, and 
 thus obtain our second axiom. 
 
 Axiom II. Figures may be moved in space without 
 c/iange of shape or size. 
 
1 6 Elements of Plane Geometry. 
 
 The full meaning of this is : 
 
 Two figures 7vhich can be made to coincide at one 
 part of space can be made to coijicide at any other part, 
 whatever be the path on which each figure is moved 
 from the former position to the latter. 
 
 This axiom implies that figures which are possible 
 at one part of space are possible at ev^ery other part ^ or 
 that space is eveiyivhei'e alike. 
 
 § 39. Definition of Congruence.— T'rt:'^ figures 
 which can be made to coi?icide are said to be congi'uent. 
 
 In many text-books on geometry such figures are called 
 equal in all respects, or identically eqnal. 
 
 Congruent figures are equal in all respects except- 
 ing in position. To decide whether two figures are 
 congruent we must try to make them coincident, or 
 we must, as this operation is called, apply the one to 
 the other. 
 
 Figures which agree in shape, but not in size, are 
 called similar, and figures which agi*ee in size, but not 
 in shape, are called equivalent. 
 
 Figures which are congruent are, therefore, similar 
 and equivalent. 
 
 In the present treatise congruent figures only will 
 be treated of. 
 
 § 40. Experience shows not only that solids may 
 be moved, but also that such motion is possible if one 
 or two points of the solid be fixed. 
 
 If a weight be suspended by a string it can swing about 
 freely. As long as the string remains stretched we may consider 
 the weight together with the string as forming a single solid of 
 which one point, the fixed end of the string, is immovable. 
 
 If the weight is suspended by two strings, like the seat of a 
 
Axiom of tJic Line. ly 
 
 swing, as long as both strings remain stretched we may again 
 consider the whole as one solid, of which two points are fixed. 
 It will still be' able to move, and, when turned quite round, 
 will come back to its first position. 
 
 Assuming this property to hold for all geometrical 
 figures, we obtain another axiom, namely, 
 
 Axiom III. A figure with tivo points fixed can 
 still he moved, but only in one way, though in either 
 sense, and 'will, if moved far enough in either sense, 
 return to its original position. 
 
 CHAPTER II. 
 
 LINES AND PLANES. 
 
 § 41. The Line. — It we suspend a weight by a 
 string, the string becomes stretched, and we say it is 
 straight, by which we mean to express that it has 
 assumed a peculiar definite shape. If we mentally 
 abstract from this string all thickness, we obtain the 
 notion of the simplest of all lines, which we call a 
 straight line. 
 
 We may suppose the string to go over a pulley, 
 and then, on allowing more string to pass over, the 
 weisrht woiild move downwards. Thus the stretched 
 
 o 
 
 part of the string would become longer and longer, 
 till the weight reached the ground. Now we may 
 suppose that the pulley is raised, or that the weight 
 sinks into a hole in the ground ; thus we might 
 lengthen, or produce the straight part of our string 
 c 
 
1 8 Elements of Plane Geometry. 
 
 both ways. And this process, though practically we 
 should soon find it impossible to go further, may be 
 conceived to be continued to an indefinite extent. If 
 now the string became rigid, so that we might take it 
 out of its original position without changing its shape 
 — that is to say, without destroying its straightness^ 
 and if at the same time the string lost all thickness, it 
 would give us the notion of a straight line. 
 
 § 42. A straight line will in future be called a line 
 simply. All other lines will be called curved lines, or 
 curves. 
 
 We have obtained the following property : — 
 
 A line is of indefinite extension — that is, a point 
 which describes it may move along it, in either sense, 
 to a greater and greater distance, until it is lost in 
 unknown regions, of which our experience does not 
 teach us anything. 
 
 The word ' line,' when used without qualification, 
 will always denote the line in its entirety, that is, ex- 
 tending indefinitely in both directions. 
 
 § 43. According to Axiom II., we may move our 
 line freely in space. We may move it so that one 
 point in it is brought to any given position, and we 
 can thus make a line pass through any point A given 
 arbitrarily in space. 
 
 If this point A remains fixed, we may still, accord- 
 ing to Axiom III., turn the line about it. The line 
 will then sweep through space, and, while always 
 passing through A, will pass through fresh points 
 besides A at every change of position ; and we can 
 easily convince ourselves that it can be made to 
 pass through any second point B, given also arbitrarily 
 
Axiom of the Line. 19 
 
 anywhere in space. At least between any two points 
 within our reach we may stretch a string, hence we 
 may draw a hne between them. 
 
 We thus come to the conclusion that a line may 
 be placed in space so that it passes through any two 
 given points, or that through any two giz'en points 
 always at least one line may he placed. 
 
 § 44. We may of course obtain as many lines 
 as we like. Suppose we take two. They may have 
 any positions in space, and do not necessarily meet. 
 But we may move them so that they y\g. 4. 
 
 both pass through a point A. In 
 this case we say that the lines ineet^ 
 or intersect, or cut each other. 
 
 We may further move both lines so that they pass 
 through two points, A and B. If this is done, our two 
 lines have two points in common. 
 
 Our notion of straightness suggests at once that 
 these two lines have not only the two points A and B 
 in common, but all others besides ; that the lines, in 
 fact, coincide throughout their length. Reference to 
 the strings, as rough representations of the lines, cuts 
 off every other assumption. For if we stretch two 
 strings, and keep them close together at two points, 
 they will fall together throughout their length. We 
 suppose that our two lines will have the same property, 
 that is, that two lines which have two points in com- 
 mon have all points in common. 
 
 § 45. We have thus obtained properties of lines by 
 abstraction from observation. If we state them in 
 form of a proposition this must be taken as an axiom. 
 
 Thus we obtain the following axiom : — 
 
20 Elements of Plane Geometry. 
 
 Axiom IV. Of the Line. Through two points 
 always one, and only one, line can be draiun. 
 
 This statement includes all our previous results 
 relating to the line. It is, in fact, only another way 
 of saying that if we draw two lines through two points, 
 these hues will fall together, so as to form only a 
 single line ; or 
 
 Two lines which have two points in commofi coin- 
 cide throughout their indejinite extension, that is to say, 
 every point in the one, both betwee?i and beyond the 
 two first points, coincides with a point of the other. 
 
 Making use of the terminology of § 39, we may say 
 that lines are congruent, and that they are coincident 
 if two points of the one coincide with two points ot 
 the other. 
 
 This includes the following theorem : 
 
 Theorem : Two different lines cannot have more 
 than one poitit in common. 
 
 Two lines have not, however, necessarily a point 
 in common. 
 
 § 46. As all lines are congruent, we may consider 
 them all as copies of one another. Thus, if we want to 
 see whether any curve is straight, that is, whether it is 
 a line or not, we try to make it coincide with a line. 
 In order to draw lines on paper we use the edge 
 of a ruler or straight-edge, and move the point of a 
 pencil along it. The pencil line thus obtained is, 
 of course, material, and has breadth and thickness, 
 however fine we draw it. It gives us, therefore, not a 
 geometrical line, but only a representative, like the 
 string which we used before as an illustration. 
 
 Geometrical lines have no thickness. Hence if we 
 
Pencil of Lines. 21 
 
 place any number of lines through two points they 
 all coincide and give only a single line, which still has 
 no thickness, however many lines are put together — 
 just as we saw before that any number of points may 
 be brought together without giving anything more than 
 a point. 
 
 If we wish to test the straightness of our straight- 
 edge, we draw by its aid a line on paper — that is, we 
 take a copy of it — and then see whether the edge of 
 the straight-edge coincides everywhere with the line 
 whenever it is made to coincide Avith it at two points. 
 The straight-edge should be tried in this w^ay against 
 each side of the line drawn. If the edge coincides 
 with the line now, it is straight, 
 
 § 47. Join of Points and of Lines. — The line join- 
 ing two points is called the join of the two points. 
 The point common to two intersecting lines is called 
 theyW;/ of the two lines. 
 
 § 48. Pencil of Lines. — A fixed point A may be 
 joined to all other points in space. We get thus all the 
 lines which can be drawn through the point A. The 
 aggregate of all these lines is called a pencil of lines, 
 or a pencil of rays, the lines being in this case often 
 called rays. The fixed point is called the centre, or 
 the base, of the pencil. Any one of these rays is said 
 to be a line in the pencil, and also to be a Ujic in the 
 fixed point. In this sense we say, not only that a 
 point may He in a line, but also that a li?te may lie 
 in a point, meaning that the line passes through the 
 point. This mode of expression, or the use of the 
 same phrases in reference to lines and points, will be 
 seen hereafter to be particularly convenient. 
 
22 Elements of Plane Geometry. 
 
 § 49. Eotation about an Axis. — According to 
 Axiom III., any figure may be moved if two points 
 of it, A and B^ be fixed. If we now take a line joining 
 AB, and turn a figure about these two points, the 
 different positions of the Une AB will all be lines pass- 
 ing through A and B^ and will therefore all coincide. 
 
 Hence, if a figure turns about two fixed points, the 
 line joining them will remain in its original position, 
 and the motion of the figure will be the same if, instead 
 of the points originally fixed, we fix any other pair of 
 points, or even all the points, in the line joining them. 
 
 This motion is called rotation^ and the fixed line 
 the axis of rotation. 
 
 § 50. The Plane. — Just as the straight line is 
 simpler than any curved line, so a flat surface is 
 simpler than any curved surface. 
 
 The notion of a flat surface we again obtain by 
 observation ; the surfaces of walls and of many pieces 
 of furniture are, roughly speaking, flat. The still 
 surface of a lake, or the surface of a well-polished 
 looking-glass or flat mirror, gives even a better idea 
 of flatness. 
 
 A flat' surface is in geometry generally called a 
 plane surface^ or a plane. 
 
 § 51. The looking-glass, as well as the lake, is 
 limited in extent, and gives us therefore the notion 
 of a plane which is also limited. But we know that 
 looking-glasses are of diff"erent size, and if we have a 
 particular one before us we may imagine a bigger one 
 without difficulty. We thus imagine a plane greater 
 than the one directly suggested by the surface of the 
 glass ; and to this our imagination there is no limit. 
 
The Plane. 23 
 
 This leads us to consider a plane as being of unliinited 
 extension in all directions. 
 
 A plane thus divides space into two parts, one part 
 on each side of it, the one below and the other above 
 the plane, or the one in front and the other behind, 
 or the one to the right and the other to the left, &c. 
 These two parts of space are completely separated 
 by the plane, so that a point which moves from one 
 side of the plane to another must necessarily pass 
 through the plane. The plane is continuons. 
 
 § 52. If we now conceive a plane, that is, a flat 
 surface of unlimited extension, we may, according to 
 Axiom IL, move it freely through space, and may do 
 this until a point on it comes to a point A which has 
 been chosen anywhere in space. It is then said that 
 the plane passes through A. If we keep A fixed we may 
 turn the plane about it, and may do so until the plane 
 comes to pass also through a second fixed point B^ 
 likewise chosen arbitrarily in space. According to 
 Axiom III. and § 49, we may still move the plane, as 
 only two points of it are fixed, by turning it about the 
 line joining them, and this turning may be continued 
 until the plane passes through a third point C, chosen 
 arbitrarily, like A and B. Then our plane ^^^ll, as a 
 rule, be fixed. Thus it appears that we may place a 
 plane so as to pass through three points, A^ B, C, 
 chosen anywhere in space. 
 
 But if C happens to lie on the line joining A and 
 B, then a plane through A and B, which did not pass 
 through C, could never be made to pass through C by 
 being rotated about A and B ; for if it did contain C 
 in one position, it would contain it in all positions, as 
 
24 Elements of Plane Geometry. 
 
 this point would remain fixed during rotation. We 
 ought, therefore, to hmit the conckision arrived at as 
 follows : 
 
 Through three points which do not lie in a line we 
 may always pass a plane. Whether a plane may be 
 drawn through three points which do lie in a line, 
 remains for the moment an open question. 
 
 § 53. If we take two flat mirrors and put their 
 polished faces one on the other, if they are well made 
 their surfaces will touch throughout — that is to say, 
 every point in the one will coincide with one point in 
 the other as far as both extend together. The one 
 may further be made to slide along the other without 
 destroying this coincidence. 
 
 This suggests a new property of planes — namely, 
 that on placing two planes one on the other, they may 
 be made to coincide ; or, that all planes are congruent. 
 
 According to § 52 any plane may be placed through 
 any three points which do not lie in a line, and then 
 its position is fixed. Since planes are congruent it 
 follows that, if we place two planes through the same 
 three points, they may coincide throughout ; and it 
 will become evident that they must do so, if we con- 
 sider the result of attempting to make three points in 
 the face of one mirror fall on the face of another mirror. 
 Experience thus leads us to the following conclusion, 
 which we state in the form of an axiom, thus : 
 
 Axiom V. Of the Plane. Through three points 
 which do not lie in a li/ie, one, and only one, plane may 
 always be drawn. 
 
 This may also be stated thus : — Tioo planes coincide 
 throughout their indefinite extension if three points in 
 
Intersection of Planes. 25 
 
 the ofie, not in the same line, coincide loith three points 
 in the other ; or thus : 
 
 Three points not in a line are necessary and sufficient 
 to detcfvnine a plane. 
 
 § 54. Intersection of two Planes. — If two planes 
 have two points, A and B, in common, they must neces- 
 sarily have more points in common. For, since each 
 extends continuously without limit, a point moving 
 in the one plane through the point A ox B will 
 cross the other plane at this point (comp. § 51) ; hence 
 one plane will lie partly on the one and partly on the 
 other side of the second plane. They must therefore 
 intersect. 
 
 The intersection of two planes must be a line, for if 
 three points common to both the planes do not lie in 
 a line, the two planes will be coincident, and this we 
 do not suppose. Hence 
 
 Theorem : The intersection of two planes is a line. 
 This line is called the join of the two planes. 
 
 .§55. As the points A and B are common to both 
 planes, it follows that the hne of intersection must be 
 the line joining AB. Hence the line AB lies in 
 each of the two planes. But the two points A and 
 B may be taken anwhere in the first plane, and thus 
 we see that 
 
 Theorem : A line luhich has two points in common 
 with a plane lies altogether in that plane. 
 
 § 56. Axial Pencil. — Through a line we can thus 
 draw an unlimited number of planes, namely, all the 
 planes which pass through two points in that line. 
 
 The aggregate of all these planes forms what is 
 called an axial pencil of planes, or an axial pencil 
 
26 Elements of Plane Geometry. 
 
 simply, the common line being the axis of this 
 pencil. 
 
 § 57. Join of Line and Plane. — A line ivhich does 
 not lie altogether in a plane cannot have more than one 
 point in common with the plane^ for if it had two in 
 common it would lie in the plane. This does not, 
 however, imply that every line has a point in common 
 with every plane, but if a line meets a plane it meets it 
 in one point only. This point is called the i^itersection 
 or t\\Qjoin of the line and plane. 
 ^^' ^' § 58. From this, again, it fol- 
 
 lows that th7'ee planes which do 
 not pass through the same line 
 cannot have more than one point 
 in conwion ; for the points com- 
 mon to two planes lie on a line, and 
 this line can have only one point 
 in common with the third plane. 
 § 59. y^ li/ie and a point luithotit it determine a plane. 
 For, any two points in the line together with the given 
 point determine one plane, which according to § 55 
 and by its construction contains the line and point. 
 
 § 60. The most important properties, as far as the 
 following investigations are concerned, are, Jirst, that 
 all planes are congruent, hence that properties proved 
 for one plane hold for all ; and secondly, that lines may 
 be drawn in a plane : or more precisely, that a line 
 which joins any two points in the plane lies altogether 
 in the plane. 
 
 It is therefore possible to draw in a plane figures 
 which consist of any number of lines. Of course curves 
 may also be drawn in a plane. 
 
Exercises. 27 
 
 Figures drawn in a plane are cdXl^d plane fgures. 
 
 § 61. The study of figures in a plane constitutes 
 plane geometf-y, and it is ^yith this branch of geometry 
 that we shall be concerned in this work. 
 
 Exercises : Draunng. 
 
 Materials. — Paper, straight-edge, and pencil. The paper 
 used ought to be moderately rough, to take the pencil-marks 
 easily. 
 
 The pencil miist be rather hard and well pointed, or better 
 sharpened on a fine file to a chisel-like edge. Such an edge 
 does not require as frequent resharpening as a point. For 
 writing in reference letters a softer pencil, pointed, may be used. 
 
 (i) Test of Straight-edge. — In order to see whether the 
 edge is straight, draw a line along it on the paper, and place 
 the edge from the other side against the line. If the edge again 
 coincides with the line, the edge is straight. 
 
 (2) Take two points A, B, and draw \he\x Join c. 
 
 The join of two points does not tenninate at the two points ; 
 hence in your drawing the join of A^ B should be produced 
 both ways beyond A and B. (A point is marked best by two 
 short strokes crossing each other, or by a prick with a needle, 
 but not by a dot.) 
 
 {2.0) Draw two lines a, b, and mark their join C. 
 
 Such a join may, of course, fall outside your piece of paper, 
 unless the lines be properly chosen. In more complicated 
 figures, such as are contained in the following exercises, it will 
 be found that it is not always easy to get all those joins of the 
 lines on the drawing-paper which are essential in the figure. 
 In these cases another trial should be made. 
 
 (The join of two lines is marked by drawing a small circle 
 round the point, not by a big dot.) 
 
 (3) Take three points A, B, C, and draw their joins two 
 and two. The figure is called a Three-point. 
 
 (3a) Take three lines <?, b, c, and mark their joins A, B, C. 
 The fiirure is called a Three-side. 
 
28 Elements of Plane Geometry. 
 
 (4) Take four points A^ B, C, D, and draw the join of any 
 pair of them. Show that there are six of these joins. The 
 figure is called a Four-point. 
 
 {\a) Take four lines a, b, c, d, and mark the join of any 
 pair of these. Count these points; there are six. The figure is 
 called a Four-side. 
 
 (5) In Ex. (4) mark the joins of any two of the six lines 
 obtained which are not the given points. 
 
 (5^;) Do the same with the lines joining the points in Ex. (4^;). 
 
 (6) Draw any figure you like consisting of points and lines, 
 and together with it a second figure, according to the following 
 law. Whenever you take in the one figure a line, take in the 
 other figure a point, and for every point in the first take a line 
 in the second. Where you take in the one 'a line and a point 
 in it,' or ' two points and their join,' take in the other ' a point 
 and a line through it,' or ' two lines and their join.' 
 
 Figures related to each other in this manner are said to be 
 reciprocal. Simple examples have been given in previous 
 exercises. 
 
 (7) Draw three lines/, q, r through a point 6". On each take 
 two points : viz. A, A' on p; B, B' on q; C, C on r. Join BC 
 by a line a, CA by a line b, AB by a line c. Similarly join 
 B'C by a', CA' by //, and A'B' by a line c'. Mark the join 
 of a and a' as F, of b and b' as Q, of c and c' as F. If your 
 drawing is accurate these three points F, Q, F must lie on a 
 line s, that is, the line FQ must pass through F. 
 
 (8) Draw the figure reciprocal [see exercise (6)] to the last. 
 
 (9) Take a line s and three points A, B, Fon it. Through A 
 draw any two lines a and a'. 
 
 Through F draw any one line /. Mark the points (<?/) and 
 (dp). Join B to (<?/) by a line b, and to (dp) by a line b'. 
 Mark the points {ab') and {db), and draw their join q. Mark the 
 point where q cuts the given line s, and call it Q. 
 
 Repeat the construction, only taking for any or all of the 
 lines a, d,p other lines through A and F. If your drawing is 
 correct the last line will again pass through Q. 
 
 (10) Draw the figure reciprocal to the last. 
 
29 
 
 DIGRESSION ON LOGIC. 
 
 [i] A Proposition is a sentence which puts before us or states 
 a fact. 
 
 Thus ' Space is a fundamental conception ' is a proposition. 
 It consists of three parts. The first is 'space,' the third is 
 'a fundamental conception.' These are connected by the 
 word ' is ' (called the copula)^ and thus the proposition is 
 formed. 
 
 The meaning of a proposition will be better understood after 
 the following remarks. 
 
 [2] We may divide all things or notions into classes, accord- 
 ing to their properties. 
 
 A class may be described by a statement of the properties 
 which enable us to decide whether a thing belongs to it or not. 
 
 vSuch a statement is called a dejiniiioji. 
 
 Very often a class has a special najiie. In fact, nouns and 
 adjectives are names of classes. Thus man is the name of a 
 class, so is white. Some classes are denoted by compound ex- 
 pressions, as ' white man, ' or ' boundary of a solid ; ' others by 
 stating their definition. We denote for the present a class by a 
 letter, such as A, B, C, or by class A, class B, class C. 
 
 [3] A class may consist of only one single individual. 
 • Queen of England ' is such a class ; ' geometry ' and ' space ' 
 are others. 
 
 [4] If we denote by A the class of things possessing any 
 given property, all other things not possessing this property form 
 another class, which is called the negative to the first, and which 
 is denoted by ' not A.' 
 
 Any one thing belongs either to the class A or to the class 
 'not A,' but no thing belongs to both. It follows that not only 
 is ' not A ' the negative of A, but that also A is the negative of 
 'not A.' 
 
 The class A and the class ' not A ' are imitiially the negatives 
 of each other, and both together include all things in the universe. 
 
 In most speculations, however, we do not treat of all things 
 in the universe, but only of those belonging to a certain class. 
 
30 Elements of Plane Geometry. 
 
 We thus treat that class as our universe, leavmg out of consider- 
 ation everything that does not belong to it. 
 
 Within this chosen class we form again the classes '^-i' and 
 ^ not A,^ and call them still mutually the negatives of each 
 other, though they do not now apply to the whole universe. 
 Thus in a treatise on colour it matters little whether or not we 
 include music, or motion, or mathematics in the class 'not red.'' 
 And in geometry, where we have to deal with geometrical 
 things only, it would be needless to state explicitly that we do 
 not include in the class * not triangle ' stars, or trees, or tele- 
 phones, or anything which is not a geometrical figure. Though 
 this is only partial negation, and ought to be distinguished as 
 such, as by speaking in the above examples of ' colour not red ' 
 and of 'figures which are not triangles,' instead of saying merely 
 'not red,' 'not triangles,' all rules given here will hold for it, 
 and it will not be necessary for our purpose to state them spe- 
 cially for such cases. But it has to be remembered that in the 
 following pages 'not A' may mean partial negation in the sense 
 just explained. 
 
 [5] A proposition may be expressed in the form 
 
 ^is^ (I), 
 
 This says that, if a thing belongs to class A, then it also 
 belongs to class B, 
 
 If this is trae, then every individual A lielongs 10 class B ; 
 hence no A belongs to the class ' not B," or no thing 'not B' is 
 a thing A ; that is, every ' not B ' is a ' not A.' 
 
 The propositions 
 
 ^ is ^ (I), 
 
 and ' not B ' is ' jiot A ' (2), 
 
 express thus the same fact, and are only different forms of the 
 same proposition. 
 
 Definition.— 7y/<' propositions (i) a)id (2) are called each the 
 contra-positive form of the other. 
 
 Though both forms express the same fact it is nevertheless 
 often of importance to consider both. The contra-positive form 
 often puts the truth expressed in a different light, so that the full 
 
Digression on Logic. 31 
 
 meaning of the statement made may be more easily compre- 
 hended. 
 
 [6] The statement A is B means that every individual 
 belonging to the class A also belongs to the class B. The class 
 B thus contains all the individuals of class A ; that is to say, 
 some of the B ^xq A. It may, however, happen not only that 
 some but that all individuals of B belong to class A. 
 
 In other words, if we know that A is B we also know that 
 
 sotiie B is A, 
 whilst it may possibly happen that 
 
 all B is A, 
 or, more shortly, that 
 
 ^ is ^ (3). 
 
 This proposition may be true, but it is not a logital conse- 
 quence of the given proposition that A is B. 
 
 Definition.— T/ie Propositions 
 
 A\%B (I) 
 and B is A (3) 
 
 are called each the converse to the other. 
 
 The contra-positirc'e of the converse (3), namely, 
 'not ^' is 'not ^' (4), 
 
 is called the olrcerse of the original proposition. 
 
 [7] From the truth of either of two converse propositions, 
 that of the other cannot be infeiTcd. If, however, both are tnie, 
 the classes A and B are identical. 
 
 If proposition (i) is true, the proof that a thing belongs to 
 class A also proves that it belongs to class B. In other words, 
 if A is B, then in order to know that a thing is B it is sufficient 
 to know that it is A, but it is not necessary that a thing must be 
 A in order to be B unless A and B are identical. 
 We thus see that the three Propositions :— 
 I ' The class A and the class B are identical ; ' 
 2. ' The proposition A is B and its converse B is A are 
 both true j ' 
 
32 Elements of Plane Geometry. 
 
 3. 'In order to prove that a thing is B it is necessary and 
 sufficient to show that it is A ' — 
 are only different forms of the same proposition. 
 
 [8] Example I. Let us take as the original proposition the 
 following from § 16 : 
 
 *The path of a moving point is a cm-ve.' The contra-posi- 
 tive is, ' A thing which is not a cm-ve is not the path of a moving 
 point. ' 
 The converse is — 
 
 * A curve is the path of a moving point,' of which the contra- 
 
 positive form is, ' Anything which is not the path of a 
 moving point is not a curve.' 
 
 In this case it follows from §§ 16, 17 that the converse propo- 
 sition is true, for a point may be moved along every curve. The 
 class 'path of a moving point ' and the class 'curve ' are identical. 
 
 If we treat in the same Avay the proposition in § 23, 
 
 ' A surface is the path of a moving curve, ' 
 the converse is — 
 
 ' The path of a moving curve is a surface,' 
 
 Here the converse proposition is not always true, for, as has 
 been pointed out in § 22, the path of a moving curve is not 
 always a surface. Hence the class ' surface ' and the class ' path 
 of a moving curve ' are not identical. 
 
 The same is true of the proposition in § 32: 
 
 'The path of a moving surface is, in general, a solid.' 
 
 [9] Example II, Let us consider the proposition Axiom I. : 
 
 ' Space is of three dimensions.' 
 
 In this form the two classes are ' space ' and ' thing of three 
 dimensions. ' 
 
 The second class may be better defined if we use the second 
 orm of the Axiom : 
 
 « Space is a three-way spread with points as elements. ' 
 The contra-positive of this is — 
 
 * Anything which is not a three-way spread with points as 
 
 elements is not space. ' 
 This includes, as a special case — 
 
 ' A three-way spread which has not points as elements is not 
 space. ' 
 
Digression on Logic. 33 
 
 The converse is — 
 
 ' A three-way spread with points as elements is space. ' 
 Or in its contra-positive form — 
 
 ' Anything which is not space is not a three-way spread with 
 points as elements. ' 
 
 This proposition is not true, for three-way spreads with 
 points as elements are conceivable with properties very different 
 •from those of space. Hence the classes 'space' and 'three-way 
 spreads with points as elements ' are not identical. 
 
 [10] We have to deal in Geometry with different kinds of pro* 
 positions, viz, with Definitions, Axioms, Theorems, Corollaries, 
 Slid Problems. What a Definition is we have seen already, viz, 
 
 A proposition "which states the pj-operties that enable us to decide 
 li'hsther a thing belongs to a class or not is called a Definition oj 
 tlie class. 
 
 A good definition must state as many properties as are suf- 
 ficient to decide whether a thing belongs to a class or not, but 
 not more than are necessary for this purpose. 
 
 [11] We have in this book only to treat of classes relating 
 to Geometry, principally of figures. Whenever we introduce a 
 new figure we have to define it. This can only be done by re- 
 ference to simpler figures out of which it is formed. These 
 simpler figures, again, are defined by aid of still simpler ones, till 
 at last we arrive at the simplest figures possible, which from 
 their very simplicity cannot be defined geometrically. These are 
 point, curve, surface, solid. We have notions of these, but we 
 cannot give true definitions of them, at least not at the beginning 
 of our study. The same holds for lific, plane, and space. We may 
 say (§ 50) *a flat surface is called a plane,' and call this a defini- 
 tion of a plane. But this defines the class ' plane ' by aid of the 
 class ' flat. ' What * flat ' is remains undecided. But then there 
 is the Axiom V. , which tells us what we do understand hy fiat. 
 It enables us, if we apply it to a surface and a copy of it, to 
 decide whether a surface is a plane ; and thus it takes the place 
 of a definition. The same is true for the other Axioms. All 
 six Axioms together define space. 
 
 The first tells us that space is a spread. To distinguish it 
 from other spreads we learn that 'x is a fhrre-icay sprerd, and 
 D 
 
34 Elements of Plane Geometry. 
 
 that it has points as elements. The second says that the three 
 ways in which we may go from one element to others are of the 
 same kind ; that they are not different, as in the example given 
 in § 28 about the two ways of changing a musical tone. The true 
 nature of the remaining axioms cannot be explained at this stage. 
 It may be here sufficient to state that many three-way spreads of 
 points are conceivable, and that the axioms state which of these 
 conceivable ones has the properties that are observed in space. 
 [12] A Theorem zs a propositioji the truth of which is iti- 
 f erred from other proposifio7ts that are kiiozvn to be true. The 
 process by tvhich this truth is established is called the Proof or the 
 Demonstration of the Theorem. 
 
 A Corollary to a Theorem is a proposition whose truth follows 
 immediately from that of the Theorem. 
 
 A Geometrical Theorem always relates to some Figure, and 
 says that a figure which has a certain property has of necessity 
 also another property. To bring it to the form of a simple 
 proposition A is B, we may state it thus : 
 
 * A figure which has a certain property' is 'a figure which 
 has another specified property.' 
 
 The first part contains the name or definition of the figure 
 to which the Theorem relates, and is called the Antecedent of 
 the Theorem. The second jDart contains another property not 
 stated in the antecedent, and is called the Consequent. 
 
 [13] In order to understand the meaning of a Theorem, the 
 student should always at first make out clearly the nature of the 
 figure about which a statement is made, and should, if no figure 
 is provided, draw one for himself. 
 
 This is advisable even if a figure is inserted in the book, for 
 nothing affords so much assistance in understanding the im- 
 portance of the different properties which a figure possesses, and 
 in realising how its different elements (lines and points) are 
 related to one another, as seeing how they come into existence 
 one after the other. The student should begin by finding out 
 which parts of the figure are given arbitrarily, that is to say, 
 which lines he may draw as he likes or which points he may 
 assume where he likes. Then follow other parts which have 
 a definite and prescribed relation to these. 
 
Digression on Logic. 35 
 
 [14] From what has been said about Propositions it follows 
 that of two Theorems which are one the contra-positive of the 
 other, only one needs 2, geometrical proof ; the other is a logical 
 consequence of it. 
 
 The converse and the obverse T/ieoj-ems, on the other hand, 
 are not logical consequences of the original, and one of them, 
 therefore, aliuays requires a special proof. This takes 
 different forms, of which the more important ones may be 
 enumerated. 
 
 1. If it has been proved that every A is B, and also that 
 there is but one individual in the class B, then we infer that 
 B is A. 
 
 The geometrical proof required for the converse is here con- 
 tained in the proof that there is but one B, This shows the 
 importance of proving that there is but one in a class. 
 
 2. If it has been proved that every A is a. B, that no two ^'s 
 are the same B, and that there are as many individuals in class 
 A as in class B. 
 
 For example, if you see ten soldiers in a room with a 
 number of swords hanging on the wall, and know that every 
 soldier owns one of the swords, then conversely you also know 
 that ever>' sword belongs to one of the soldiers, if you find on 
 counting that there are but ten swords. 
 
 3. In many cases the converse theorem or its contra-positive, 
 the obverse of the original, has to be proved quite independently 
 of the original theorem. 
 
 [15] In a Geometrical Problem it is generally required not 
 to find or prove properties of a given figure, but to find a figure 
 which shall have certain prescribed properties. 
 
 The determination or consti-uction of such a figure is called 
 the Solution of the problem. 
 
 When the solution has been obtained, a Proof is required that 
 the solution is correct. This proof in many cases is contained 
 in the reasoning which has led to the solution. In other cases 
 the solution has been obtained by intuition or by a happy guess, 
 and in these cases the /;w/ must not be omitted. 
 
 Exercise. — Determine the logical connexion between Axiom 
 v., § 53, and the theorems in §§ 54, 55, and 57. 
 D 2 
 
36 Elements of Plane Geometry. 
 
 CHAPTER III. 
 
 SEGMENTS AND ANGLES. 
 
 § 62. Reciprocal Figures. — Every figure in a plane 
 consists of a combination of points and lines, and of 
 curves generated by the motion of a point or by the 
 motion of a line. Points and lines are, therefore, 
 called the elements of plane figures. 
 
 At first sight it might appear that points only 
 deserve this designation, but we shall soon see that 
 lines are of equal importance in the generation of 
 figures. In many cases we consider points only as 
 elements, in others lines only, in others again both 
 points and lines. In most cases we can, when one 
 figure is given, construct another such that lines take 
 the place of points in the first, and points the place of 
 lines. Any theorem concerning the first thus gives 
 rise to a corresponding theorem concerning the second 
 figure. Figures and theorems related in this manner 
 are called reciprocal figures or reciprocal theorems. 
 
 §. 63. Let us suppose, as a simple example, a 
 figure consisting of two points, 
 ^'^' ^' ^ 4, B (fig. 6) ; the reciprocal 
 
 ' ~^ figure will then consist of two 
 
 lines, ^, b (fig. 7). "The first figure has the property 
 Pjf;_ ^_ that there is one line, r, joining 
 
 the two points. Correspond- 
 ing to this we find in the se- 
 cond figure the property that 
 two lines a^ b have a point C 
 in common. Hence we get as propositions, corre- 
 sponding in the manner indicated, the following : 
 
Reciprocal Figures. 37 
 
 Two points have a line in common ; or, the Join of 
 two points is a line. And reciprocally — 
 
 Two lines have a poifit in common ; or, the join of 
 two lines is a point. 
 
 § 64. The first of these propositions is always 
 true ; the second, though it is generally true, is not 
 always so, for we shall soon see that two lines in a 
 plane do not necessarily intersect. This, however, is 
 an exception, but it has to be remembered. 
 
 The fact that a proposition is tnie as a rule, but with some 
 exceptions, is expressed by the words ' in general. ' Hence when- 
 ever these occur in any proposition it is to be understood that 
 there are exceptions to the proposition, but so that among an 
 infinite number of cases in which the proposition is applicable 
 there are only ?l finite number of exceptions. The last proposi- 
 tion ought, therefore, to be stated more exactly : 
 
 Two lines have in general a point in common^ and, 
 if two lines have a join., this is a point. 
 
 § 65. In all that follows points will be denoted 
 by capital letters, A^ B^ C . . . F, Q . . . and lines by 
 small letters a, b, c . . . p, q . .. The join of two 
 elements will be denoted by putting the letters, indi- 
 cating the elements, together. Thus, the line joining 
 the points A and B is called the line AB., whilst 
 ab denotes the point of intersection of the lines 
 a and b. 
 
 § (id. Row of Points, Pencil of Lines. — A line 
 contains an infinite number of points, which, accord- 
 ing to § 19, form a ro7u of points on the line ) but, for 
 shortness sake, this- will be called in future simply a 
 ro7u of points. 
 
 The line containing the points is called the base 
 of the row. 
 
38 
 
 Elements of Plane Geometry. 
 
 As the figure reciprocal to a row, viz. all points 
 in a line, we get all lines in a point, or, in common 
 language, all lines passing through a point. 
 
 The aggregate of all lines in 
 a plane which pass through a 
 given point is called a pencil of ' 
 lines in a plane, or a flat pencil, 
 to distinguish it from the pencil 
 of lines in space (§ 48). In 
 plane geometry, where only figures 
 in a plane are considered, it is 
 sufficient to call it simply a pencil 
 of lines, or a pencil of rays. 
 
 The common point is called the centre or the base 
 of the pencil. 
 
 We have thus the reciprocal propositions 
 A point moving along a line describes a row. 
 A line turning about a point describes a pencil. 
 In either case the motion, or turning, may be 
 effected in one or the opposite sense. (§ 17.) 
 
 Sometimes it is convenient to consider each line 
 in the pencil as terminated at the centre, and as 
 described by a point moving from the centre in one 
 way to an indefinite distance. The line is then called 
 a half -ray. 
 
 § 67. Direction. — Two lines or half-rays drawn 
 through the same point are said to have the same 
 direction if they coincide ; otherwise they are said to 
 have different directions. Two half-rays, of which 
 one is the continuation of the othfer, are said to have 
 opposite directions. 
 
 Two half-rays which have opposite directions lie, 
 
Segment of a Line, 39 
 
 therefore, in the same hne, and are of opposite sense 
 in this line. 
 
 § (i%. Segment. — If two points, A^ B, be taken in 
 a line, then that part of the line which is bounded by 
 A and B is called (§ 21) a segment of the line, or where 
 no ambiguity is possible, simply a segment. The two 
 points A and B are called the end points of the seg- 
 ment. The segment is denoted by AB. 
 
 § 69. Two segments, AB and CD, can always be 
 placed one on the other in such a manner that C falls 
 on A, that the t^vo lines of w^hich they are segments 
 coincide, and that B and D lie on the same side of A. 
 In this position the point D falls either between A and 
 B, or on B, or beyond B. A point moving from A 
 along the line wall either first reach D, and afterwards 
 B, or it will reach B and D at the same time if these 
 coincide, or it will reach first B and then D. 
 
 In the first case it is said that CD is less than A B, 
 in the second that CD is equal to, or of the same 
 length as, AB, and in the third that CD is greater 
 than A B. This is expressed in symbols as follows : 
 
 CD < AB means CD is less than A B, 
 CD = AB „ CD is equal to AB, 
 CD > AB „ CD is greater than AB. 
 
 These expressions relate to the kngt/i of the 
 segments. 
 
 § 70. Measurement of Segments. — ^^^len any two 
 segments are given, one, and on/y one, of these three 
 relations must necessarily hold good. To decide 
 which exists, the above criterion requires that w^e 
 place the one segment on the other. This operation 
 
40 Elements of Plane Geometry,. 
 
 is practically seldom possible. If, for instance, both 
 segments are drawn on the same paper, we cannot 
 actually move the one line towards the other unless 
 we cut the paper. We must, therefore, use a different 
 method. This consists in taking a third segment? 
 PQ,^ movable in space, which we compare first with 
 the one, AB, and afterwards with the other, CD. 
 If we take this third segment P Q equal to A B^ and 
 find on moving it to CD that it is also equal to CZ>, 
 we conclude that CD equals A B. For if we sup- 
 pose that the segment PQ is made to coincide with 
 CD, and if we move both together to A B, keeping 
 them coincident, then on making P Q coincident 
 with AB, to which it was supposed equal, CD will 
 also coincide with A B. (Axiom II. § 38.) 
 
 This is generally expressed by saying, two magni- 
 tudes w/ii'c/i are each equal to a third are equal to one 
 another) and the statement is taken as an axiom. 
 We see that it is in the present case a consequence 
 of our Axiom II., which relates to the movability of 
 figures without change of shape or size. It may also 
 be taken to be a definition of equality of length of 
 segments, or rather to be a criterion. 
 
 All our measurements of length depend upon this 
 proposition. 
 
 § 71. For geometrical purposes this is done 
 generally by aid of a pair of compasses. Their two 
 points, which we may call P and Q, can within certain 
 limits be opened to any distance required, so that they 
 can be made to coincide with the end points of the 
 segment ^^, and then a segment P Q, equal in length 
 to A B, may be carried a])out and compared with the 
 segment CD. 
 
Equal Segments. 41 
 
 A pair of compasses may thus be said to be an 
 Instrument for carrying distances, or segments of 
 given length, about in space. 
 
 § 72. A segment having the end points A and B 
 may be supposed described by a point moving either 
 from A to B or in the opposite sense from B to A. 
 These two segments, which are equal in length but 
 of opposite se/ise, are distinguished d.s AB and BA 
 respectively, so t/iai AB means the segment deseril^cd 
 by a point moving from A to B. 
 
 § 73. If two different segments, AB and CD, lie 
 in the same line p, they may be compared as to sense. 
 They have the same or opposite sense, according as 
 points moving from A to B and from C to D respec- 
 tively move in the same or in opposite sense in the 
 Hne /. Or we may compare them thus : 
 
 We suppose the segment CD to slide along the 
 line/, till C coincides with A \ then D will fall either 
 at D' , on the same side of ^ y\g. 9 
 
 as B (fig. 9), or at D" on the As c- d 
 opposite side of y^ (fig. 10). d' 
 
 In the first case the two Fig. 10. 
 
 segments are of the same A. b p c 
 
 sense, in the other case they ^^ 
 are of opposite sense. 
 
 § 74. Two segments In the same line which are 
 equal in length and of the same sense will in future 
 be called equal simply, while, if they are equal in length 
 and of opposite sense, they will be called equal and 
 opposite. 
 
 It is convenient to Indicate the sense also by a 
 sign. For this purpose the signs + (read plus) and 
 
42 Elements of Plane Geometry. 
 
 — (read minus) are used, which are borrowed from 
 algebra. \i AB and CD are equal in length and of 
 the same sense, or simply equal, this is indicated by 
 writing 
 
 j^AB=^CD', 
 
 but \i AB and CD are equal in length and of oppo- 
 site sense, we write 
 
 ■\-AB=z-CD. 
 
 Generally the sign + is omitted, so that the equation 
 
 AB^ CD 
 
 indicates not only that the two segments are equal in 
 length, but also of the same sense, whilst 
 AB^-^CD 
 
 means that AB and CD are segments equal in 
 length but opposite in sense. This extension of the 
 meaning of the sign = must be kept well in mind. 
 At present this distinction of sense has a meaning 
 only if both segments lie in the same line. 
 
 § 75. The symbol AB thus means the segment 
 described by a point moving from A to B, or A B is 
 the result obtained by moving a point from A to B. 
 —AB, on the other hand, means the same segment 
 described in the opposite sense, that is, described by 
 a point moving from B to A, so that 
 
 BA=-AB. 
 This equation expresses the fact that the two segments 
 A B and BA are equal in length but opposite in sense. 
 
 Theorem : If the order of the two letters indtcating 
 the end points of a segment be changed, the segment 
 changes its sense. 
 
Sinn of Segments. 43 
 
 § 76. Sum of Segments.— If A, B, C be three 
 points in a line, then the segnient AB vs, obtained by- 
 moving a point from A to B^ and the segment B Chy 
 moving a point from B to C. If both these operations 
 be performed in succession, we obtain a segment de- 
 scribed by moving a point first from A to B and then 
 from B to C, which is equivalent to moving a point 
 from A to C, describing the segment A C. This is 
 expressed by saying A C is the sum of AB and B C ; 
 in symbols 
 
 AB+BC=AC. 
 
 This definition of the sum of two segments does 
 not suppose any definite order for 
 the points A, B, C. It holds equally ^ b c 
 in the case where B lies between A """ ' ^ 
 and C (fig. 11), in which case the moving point de- 
 scribes both segments A B and B C 
 in the same sense j or where B lies 
 beyond C (fig. 12); or where B and ^ 
 CHe on opposite sides of ^ (fig. 13). In the latter 
 cases the moving point describes the 
 second segment in a sense opposite 
 to that of the first. But the above -< < ^- 
 
 FlG. 12. 
 
 A c u 
 
 — I i- 
 
 equation holds always. 
 
 § 77. If A,B, C,D...G, If are points in a line, 
 we get in the same manner 
 
 AB + BC+CI?+...'^GIf=AJI. 
 
 This means, if a point moves from A to B to C to 
 ...to GtoH, it is at the distance of AH from the 
 starting-point A, whatever the positions of the points. 
 
 § 78. If we take the case where the last point 
 
44 Elements of Plane Geometry. 
 
 coincides with the first, the sum is equal to A A. But 
 A A has no length. This is expressed by saying A A 
 equals zero : 
 
 AA=o. 
 
 Applying this we get the important formulae 
 AB-\BA=o, 
 AB-^BC-^CA^o, 
 AB-\-BC+Cn-\-...-YGH+IfA = o, 
 which hold for any points in a line. 
 
 This means, if a point moves from A to B to C to 
 &c., and ultimately back to ^, it is at no distance 
 from A. 
 
 § 79. Having thus defined the sum of consecutive 
 segments in a Hue, we may now define the sum of any 
 two segments in the same line. 
 
 We say that A B is the sum of two segments CD and 
 E Em the sa7ne line: if a point 
 
 Fig. 14. ' J Jr 
 
 ^, „ „ ^ F can be foicnd' such that A F 
 
 ' ' ' ' ^ and FB are equal respectively 
 
 D c B p J^ ^^ ^^ ^^^^ ^^^ taking, of 
 
 F E course, account of the sense. 
 
 In symbols 
 
 AB^CD-^EF, 
 
 if a point F can be found such that A F= CD and 
 FB=EE. Similarly for the sum of more than two 
 segments. 
 
 § 80. Difference of Segments. — By the difference of 
 ttvo segttients AB and CD, we mean a segment EE, 
 7vhich, when added to CD, gives AB as sum. 
 
 Denoting the difference of AB and C D hy 
 AB—CDf\fQ have 
 
Difference of Scgnients. 45 
 
 AB-CD=EF if EF-\-CD=:AB. 
 
 From this, since AB-\-B C=AC, it follows at once 
 that 
 
 AC-BC=AB, 
 
 and also that A C~AB=B C. 
 
 In words : 
 
 The difference behueen two segments with co7fwwn 
 end-point equals the segment described by a point moving 
 from the initial point of the first to that of tht second; 
 
 The difiereiice between ttuo segments with common 
 iniiial point equals the segmetit described by inoving a 
 point from the efid point of the second to that of the first. 
 
 As a special case we get 
 
 AC-AC=o, 
 
 for it is equal to ^^. 
 
 The differefice of equal segments is zero. 
 
 § 81. We have obtained the two equations 
 
 A C+ CB=AB and A C-B C=AB. 
 Hence A C-B C=A C+ CB. 
 
 But CB and B C denote the same segment taken in 
 an opposite sense. If we use the sign — , as before, 
 to indicate negative sense, so that B C= — CB, we 
 may ^^Tite the above equation 
 
 AC-{-CB)=AC+CB, 
 
 Hence the proposition 
 
 Instead of sjibtracting a seg?nent we may change its 
 sense and add it. 
 
46 Elements of Plane Geometry. 
 
 § 82. By definition AB— CD is a segment which, 
 when added to CZ>, gives A B^ or 
 
 AB-CD^CD=AB. 
 
 If we write here -DC for CD and ^DC for 
 — CD^ we get 
 
 Here D C may be any segment whatsoever. 
 Tliis shows that if the same segment be first added 
 to, and then subtracted from, or first sjibtracted fro7n 
 and then added to, any given segment, we obtain the 
 original segment ; or addition and subtraction, as defined 
 in the above, are operations which are opposites to one 
 another : the one undoes what the other does. 
 
 § 83. The results obtained in § 82 show the 
 complete analogy between this addition and subtrac- 
 tion of segments, with the corresponding operations 
 in algebra, and also the analogy between the sign of 
 a number in algebra and the sense of a segment in 
 geometry, thus justifying the use of the same symbols. ^ 
 
 § 84. Angles. — Having thus treated of segments, 
 we have next to investigate the corresponding proper- 
 ties of the figure reciprocal to a segment. A segment 
 was defined in § 68 as a part of a row, described by the 
 Fig 15. motion of a point from one position 
 
 I A lo another position B. The 
 figure reciprocal to a row is a pencil 
 (§ dC). Hence as the figure recip- 
 rocal to a segment we get a part of 
 a pencil, described by the turning about the centre C 
 of a line in the pencil from one position a to another 
 position b. The result we call an angle. 
 
A ngles. 47 
 
 It will, however, be convenient to consider at 
 first a pencil of half-rays only. We 
 then get the definition : 
 
 Definition of Angles. — The part of 
 a pencil of /lalf -rays, described by a half- 
 ray on turning about its end point C ^ "' 
 
 from one positio7i a to another position b, is called an 
 angle. The centi-e C of the pencil is called the vertex, 
 and thefii'st and last positions a and b of the describing 
 ray are the limits of the angle. 
 
 The angle itself is denoted by a b, or, if A and B 
 are points on the limits a and b respectively, by A. CB, 
 or, where necessary to avoid ambiguity, hy ^ab or 
 Z. A CB, using the symbol /. to denote an angle. 
 
 § 85. This definition is not yet sufficient. For if 
 the limits a and b are given the angle ab h not 
 uniquely determined. We may turn ^^^ 
 a half-ray from ^ to ^ either in the 
 sense indicated by the curved arrow 
 in fig. 16, or in the opposite sense, 
 as in fig. 17. 
 
 We must therefore, besides the limits ab^ know 
 the sense of turning. It will be convenient to fix 
 once for all that sense of turning which we consider as 
 positive, and which is meant if nothing else is stated. 
 This is best done by referring to the well-known turn- 
 ing of the hands of a watch. 
 
 The turning of a 7'ay in a sense opposite to that of 
 the hands of a watch will in future be taken as positive. 
 
 The hands of a watch, therefore, turn in the negative 
 sense. Their motion was originally made to agree with 
 the apparent motion of the sun as seen by the inhabit- 
 
48 Elements of Plane Geometry. 
 
 ants of the northern hemisphere. Our positive sense 
 of turning agrees, therefore, with the sense in which 
 the earth revolves about its axis as seen from the 
 North Pole. 
 
 § 86. Angles of Continuation and of Rotation. — 
 
 Pj(, j3_ If a half-ray is turned about the 
 
 /^^ point C in the positive sense, 
 
 —T, ?? Ti from an initial position a till it 
 
 cgincides with the continuation boia beyond C, the angle 
 
 abx?, called an angle of conti7iuation. If we turn still 
 
 Pj3 further, the moving ray will ultimately coin- 
 
 a cide with a (fig. 19). In this case the 
 
 ©- 
 
 2 moving ray has made a full rotation, and 
 the angle generated is called an angle 
 of rotation. 
 
 § 87. Equal Angles. — Two angles are called equal 
 if they can be placed in such a position that their limits, 
 and therefore their vertices, coincide, and that both are 
 described simultaneously by the turning of the same 
 half-ray about their common vertex. This implies 
 that both angles have the same sense. If the angles 
 are of opposite sense whilst their limits coincide, they 
 are said to be equal in magnitude but opposite in sense. 
 Hence the two andes ab with vertex C 
 
 Fig. 20. - , . , ^, , -r 
 
 f, and cd with vertex c , are equal if, on 
 y^ placing C on C and ^ on ^ (fig. 20), the 
 
 c/'^\ line d falls on b, and both have the 
 
 ^ same sense. If, however, after placing 
 
 e on a, the moving ray arrives first at b 
 and afterwards at d, the angle ed is said 
 to be greater than the angle ab ; or, in 
 symbols, ed>a b or ab< cd, (Compare § 69.) 
 
Equality of Angles. 
 
 49 
 
 Fig. 
 
 § 88. As special cases we get : 
 
 Theorem : All angles of co7itmuation are equal, or 
 equal and opposite. 
 
 All angles of rotation are equal, or equal and opposite. 
 
 § 89. We may turn a half-ray from an initial posi- 
 tion a about C first through an angle of 
 rotation, and then further on till it 
 comes to b. Hence the angle ab \s 
 still undetermined. It may either mean 
 the angle obtained by turning from a 
 to b, or we may first turn through an angle of rotation 
 and then stop at b, or we may first turn through two 
 or three or any number of angles of rotation and 
 ultimately stop at b. Hence, there is an indefinite 
 number of angles which have a as the first and b as 
 the second limit, and which are all described in the 
 same sense. Thus a a may mean an angle of rotation 
 or an angle of no magnitude, equal to zero. 
 
 In some parts of mathematics, for instance, in trigo- 
 nometry or in apphed mathe- p.^ „3_ 
 matics, it is of importance to 
 consider all these angles. A 
 simple example \\\\\ show this. 
 
 Suppose we have a common 
 right-handed screw, arranged as 
 in fig. 22, with an arm attached 
 to its head, and we turn this 
 arm in the positive sense: the 
 nut will move downwards, and 
 this motion of the nut will be 
 the greater the greater the 
 angle through which the arm has been turned. We 
 
50 Elements of Plane Geometry. 
 
 see at once that it is a different thing turning through 
 an angle, say, of continuation, or through the same 
 angle together with one, two, or more angles of rotation. 
 § 90. Adjacent Angles. — If from a point C 
 Fig. 23. three rays are drawn ^, ^, c we 
 
 have the angles ab^ be, ac. Of 
 these any two have a limit in com- 
 & mon. Such angles are called adja- 
 cent angles. Thus ab and be, ot ac 
 c * and cb, or be and ca, are adjacent 
 
 angles. Further, ac is called the sum of the angles 
 ab and be; in symbols 
 
 ab-\-bc=^ac. 
 
 This means, as in the case of a segment, that // is 
 the same thing whether we turn first from a to b, and 
 then from b to c, or whether we turn at once from a to c. 
 
 If c coincides with a we get 
 
 ab + ba=-aa=- angle of rotation. 
 
 In words : If a ray turns fii'st fivm a to b, and then 
 in the same sense from b to a, it coifies back to its original 
 Position, having completed a full rotation. 
 
 § 91. Negative Angles. — If we further agree that 
 — ab shall mean the angle which we get by turning 
 from ^ to ^ in the negative sense, we have 
 ab + {—ab):=.aa-=^o. 
 
 But if, in the last formula of § 90, we substitute for the 
 angle of rotation a a the angle zero we have 
 
 ab + ba = o. 
 
 A comparison of these two formulae shows that 
 
 I 
 
General Definition of Angles. 51 
 
 III all investigatio7is where an angle of rotation may 
 be replaced by an angle zero lue harem 
 — ab^=ba, aa=.o. 
 
 § 92. In elementary geo- Fig. 24. 
 
 metry it is in general sufficient 
 to consider angles which are not 
 greater than an angle of rotation. 
 In future we shall, therefore, un- 
 derstand, if nothing else is stated, ^ ^ 
 by the angle ab, the angle gene- 
 rated by turning a 7' ay in the ^\^ 
 positive sense, about the vertex, 
 from a till it comes to hfor the first time. (See fig. 24.) 
 
 § 93. This definition will still apply if a and b are 
 lines unlimited in both directions, which meet each 
 other at some point C, instead of being half-rays dra-vvn 
 from that point, provided we give each line a sense, 
 which may be indicated where necessary by an arrow- 
 head. 
 
 DeMition.— By the angle a,h ^^^ 
 
 will be understood the angle in- "' y,^ 
 
 eluded by the two half -rays a and ^^ 
 
 b, which are drawn from C i?i y\^ 
 
 the positive sense of the lines a 
 and b, or the angle A CB, as 
 indicated in the figures. If we 
 change the sense of o?ie of the 
 Hnes the angle changes its value 
 
 (%-25). z' 
 
 The angle which is obtained by changing the sense 
 of both lines is said to be vertically opposite to the first 
 
 angle (fig. 26). 
 
 K 2 
 
52 
 
 Elements of Plane Geometry, 
 
 Vertically opposite angles are such that the limits of 
 the one are the continuations of the limits of the other. 
 Fig. 26. § 94. With this definition of 
 
 an angle we obtain for angles a 
 series of formulae exactly corre- 
 sponding with those obtained 
 in §§ 75-78 for segments, pro- 
 vided we always consider the 
 angle of rotation equal to zero. 
 But it must be kept in mind 
 that the angle zero may, in 
 special cases, mean a positive 
 or negative angle of one or two 
 or any number of rotations. 
 We have always 
 
 aa-=o^ ab^=^~ba, 
 ab-\-bar=o, 
 ab-{-bc+ca—o, 
 ab ^bc=^ac, 
 
 ab + bc+ . . . + ef+fg—ag, 
 ab-\-bc+ ...e/-^/a=o. 
 
 Further, if the difference of two angles be defined, 
 like the difference of segments (§ 80), as that angle 
 which added to the second gives the first, we have 
 
 ab—cb=ac, 
 ab—ac=c'b. 
 
 § 95. In some cases we must 
 retain the angle of rotation. Thus 
 if we draw from a point Ca num- 
 ber of half-rays, ^, b, c, d, we have 
 ab + be -'t-cd-\-da=^ angle of rota- 
 tion. 
 
 Fig. 27, 
 
Exercises, 53 
 
 In words, the sum of all the angles mto 7vhich 
 a pencil of half -rays is divided by a 7iumber of these 
 rays equals an angle of rotation. 
 
 Here it would, of course, be absurd to say the 
 sum is zero. But whenever we have to deal merely 
 with the relative position of the first and last limit in 
 a sum of angles, without considering the amount of 
 turning required to describe the angle, we may say, 
 aa=o, or aa=a.ng[e of rotation, or ^c7=any number 
 of angles of rotation. For either statement involves 
 the assertion that the two limits coincide. 
 
 Exercises. 
 
 (1) lfAB=2, AC=s, find BC. 
 liAB = 2, CA = z, find B C. 
 
 (2) If AB=4, AC=7, AD=-3, AE = 2, find B C, 
 BDy BE, CD, CE, and DE. Verify the results by trying 
 whether BC+CD + DB = o. 
 
 (3) Points A, B, C . . , on a line are often determined by 
 their distances OA, OB... from a fixed point (called the 
 zero-point or the origin). Prove 
 
 [a) That always 
 
 AB =0B -OA. 
 
 (/') If we wish to reckon the distances of the points from a 
 new origin Q, that 
 
 QA = OA-0\Q. 
 
 (4) The positions of lines in a pencil are often fixed by the 
 angles they make with a fixed line (called the initial line). State 
 and prove in this case the fonnulae reciprocal to those given 
 in the last exercise about points in a row. 
 
 (5) A half- ray turns about its end point first in the positive 
 sense through two angles of rotation, then through \ angle of 
 continuation in the negative sense, through an angle of rotation 
 in the positive sense, through | angl^ of rotation in the posj- 
 
54 Elements of Plane Geometry. 
 
 tive sense, and lastly through | of an angle of continuation 
 in the negative sense, it being taken for granted that an angle of 
 continuation is equal to half an angle of rotation (§ 104). 
 
 What will be the value of the angle according to the defi- 
 nition in § 92 between the original position of the ray and its 
 position after each successive step of turning ? 
 
 DRAWING. 
 
 We shall now add to our drawing instruments a pair of 
 compasses and a ' scale. ' 
 
 A pair of compasses serves to carry distances about (§ 71). 
 It is supposed at present that none of its points is replaced by 
 a pencil. 
 
 A scale serves to measure off distances in terms of a definite 
 unit. 
 
 (6) On a line set off to scale, beginning at any point, A 
 distances. 
 
 AB = + 2, units of length ; ^ C = + 5, CD =-T, 
 DE = -g, £F = + iS units of length. 
 
 The positive numbers, marked + , are to be set off in the 
 positive sense ; the negative numbers, marked — , in the ne- 
 gative sense. 
 
 \Neh^\eAB-i-BC+CD + D£ + EF= A F ; hence 
 
 3+5-7-9-15= +;• 
 
 That is, the point F ought to be, at the distance of 7 units, 
 in the positive sense from A. 
 
 Measure A Fin order to see whether your drawing is correct. 
 
 Find also, by calculation and by measuring, the distances 
 AD, BF, CF, in order to check your calculation, 
 
 (7) Draw the figures to the first four Exercises. 
 
55 
 
 CHAPTER IV. 
 
 BISECTORS OF SEGMENTS AND ANGLES. 
 
 -A point M which 
 
 Fig. 28. 
 
 § 96. Bisector of a Segment. 
 
 divides a segment A B into two 
 equal parts, A MsiYid MB, so that ^ 
 A M=MB, or MA = - AfB, '' ' ' 
 
 is called the d/sedor, or the mid point of the seg- 
 ment A B. 
 
 § 97. Bisector of an Angle. — 
 A line ;;/ through the vertex of an 
 angle a b, which divides that angle 
 into two equal parts, am and mb, 
 so that 
 
 a7n = mb, or ma ^^ — mb, 
 is called the bisector or mid ray "^ 
 
 of that angle. 
 
 § 98. The mid point M of the segment A B has 
 the property that if the segment MB be turned about 
 it through an angle of continuation, then B will fall on 
 A. If the whole segment AB he thus turned about 
 M, then the points A and B will interchange their 
 positions. 
 
 This is expressed by saying that the two points 
 A and B lie symmetrically with regard to M, and J/ is 
 called a centre of symmetry of the figure AB. 
 
 § 99. The mid ray m of an angle ab has the 
 property that if the plane of the angle be doubled 
 over by folding it along w, and turning the one angle 
 mb about m till it falls into the plane of ma, then its 
 limit b will coincide with a. 
 
56 Elefucnts of Plane Geometry. 
 
 If the whole angle ab h^ turned about m till it 
 comes back into the original plane, the two limits a 
 and b will have interchanged their positions. 
 
 This is expressed by saying that the two lines a 
 and b lie symmetrically with regard to w, and m is 
 called an axis of syinmetry of the figure ab. 
 
 § I GO. These properties of the bisectors of seg- 
 ments and angles may be used practically for finding 
 them. Thus, if a segment is given on a straight edge 
 of a piece of paper, we may bend the paper in such a 
 manner that the two end points coincide, and then 
 fold down the paper. The crease formed will bisect 
 the edge. 
 
 Similarly if a piece of paper on which we may 
 suppose an angle given be cut along the limits of the 
 angle, we may, by bending the paper, make the two 
 limits coincide and fold the paper down. The crease 
 hereby formed will be the bisector of the angle. 
 
 § loi. These operations can, of course, not 
 always be performed, if the figures are drawn on a 
 solid which cannot be bent over. But farther on we 
 shall obtain other modes of determining the bisectors 
 of segments and angles. The operations described 
 are, however, always conceivable. 
 
 We conclude that every segment and every angle 
 Fic, 3Q_ has a bisector. Furthermore, it 
 
 A T.1 B has only one bisector. For if 
 
 -I 1 — I 1- J 
 
 ^ we suppose a point to be moved 
 
 from the mid point M of a segment AB (fig. 30) 
 towards ^, say to C, then A C will be greater than 
 CB\ hence the point C cannot bisect the segment- 
 Similarly in the case of an angle (fig. 31). Hence 
 
Bisectors of Sco;iucufs and Auo;1cs. 
 
 57 
 
 Theorem : Every seg}?ie7it has one, and only 07te, 
 mid point. Every angle has one, and only on£, mid ray. 
 
 § 102. The reasoning used in Ftg. 31. 
 
 the last article gives rise to the 
 following propositions, which will 
 be found of use hereafter: 
 
 Theorem : The bisector of the 
 Sinn of two segments or angles which 
 have the sa?ne sense lies within the ^ 
 greater segme7it or angle {provided 
 that the siim of the adjacent angles is 
 less than an angle of rotatiofi) ; 
 and, conversely ^ 
 
 Theorem : Of two adjaceiit 
 seg?nents or angles of like sense, 
 that is the gj'cater li'hich contains 
 the bisector of their swn. 
 
 § 103. The above proof for 
 the existence of a bisector of a segment is derived from 
 intuition. But our intuition is limited. A segment must, 
 therefore, have both end points at a finite distance. If 
 we suppose the point describing a segment to move to 
 an indefinite distance, we should obtain a segment of 
 indefinite length, and this has no bisector. For we 
 cannot any longer conceive the segment folded over 
 so as to make its end parts coincident. 
 
 An angle, as defined in § 92, lias always a 
 bisector. 
 
 We thus see that, although there exist many 
 points of resemblance between segments and angles, 
 there are also important differences. A point in 
 ^ For the meaning of • converse' see p. 31. 
 
58 Elements of Plane Geometry. 
 
 describing a segment may move to an indefinite 
 distance, thus describing a segment of indefinite, 
 length. An angle, on the other hand, as it has been 
 defined in § 92, is always finite. To obtain an 
 angle of indefinite magnitude we must turn the de- 
 scribing line an indefinite number of times about 
 the vertex of the angle, thus taking an unlimited 
 number of angles of rotation. 
 
 Another difference is that we have among all 
 angles one of definite magnitude, viz. the angle of rota- 
 tion, whilst there is no segment similarly distinguished 
 from others. This gives rise to a series of propositions 
 about angles to which no propositions about segments 
 are reciprocal. These have now to be considered. 
 
 § 104. Right Angles. — The bisector of an angle of 
 rotation is the continuation of either of the coincident 
 limits. 
 
 An angle of continuation is equal to half an angle 
 of rotation. 
 
 If we bisect an angle of continuation a a' with A 
 Fig. 32. as vertex (fig. 32), we get two 
 
 '^ equal angles, which are called 
 
 right angles. 
 
 A 7'ight angle is half an 
 angle of continuation. 
 ■^ As all angles of continuation 
 
 are equal, it follows. 
 
 Theorem : All right angles are equal in magnitude. 
 They may of course be of opposite sense. 
 
 § 105. Perpendiculars. — Two lines which include 
 a right angle are said to \)Q perpendicular to one another. 
 If we produce the bisector m of an angle a a' of 
 
 n\ 
 
Perpendiculars. 
 
 59 
 
 this continuation ?n' 
 
 Fig. 33. 
 
 continuation beyond its vertex, 
 will bisect the angle of con- 
 tinuation a' a on the other side 
 of the line, as is seen at once 
 by folding over along ;;/. We 
 have, then, two lines intersecting 
 at A^ making all four angles 
 equal, and therefore all four 
 angles right angles. It follows 
 that all four angles contained by 
 two imlimited straight lines are 
 right angles if a?iy one of them is a right angle. The 
 tivo lines are then said to be perpendicular to one 
 another. 
 
 Consequently, through a point on a line always 
 one, and only one, line can be drawn perpendicular to 
 the given li?ie, for there is one, and only one, bisector 
 of the angle of continuation, which has its limits in 
 the given line and its vertex at the given point. 
 
 The line perpendicular to, and passing through the 
 midpoint of a segment is called the perpendicular 
 BISECTOR of the segnmit. 
 
 § 106. Also through a point 
 not in the line one perpen- 
 dicular to the line can be 
 drawn. To find it, we need 
 only fold the plane over in 
 such a manner that the crease 
 passes through the given point, 
 and so that the line falls on 
 itself again. Or we may proceed 
 thus : Fold the plane over along the given line; then the 
 
 Fig. 
 
 34- 
 
6o Elcmeuts of Plane Geometry. 
 
 point A will fall on some point A' on the other side of 
 the line. Now turn back and join ^ to ^'. The joining 
 line cuts the given line at a point B. If we again fold 
 the plane over, B remains where it is and A falls on A'. 
 Hence the two adjacent angles at B^ on the same 
 side of AA\ are equal, and therefore each equal to 
 half an angle of continuation. 
 
 This can be done in one way ojily, and therefore 
 one such perpendicular only can be drawn. For if 
 we suppose any perpendicular, A B, drawn from A to 
 the line a, and produce it beyond B, the angles ABC 
 and CBA' will be equal, as the first is a right angle. 
 Therefore on folding over along B C the line A B will 
 fall on its production, and A on some point A', which 
 must be the same point as before. Hence the per- 
 pendicular found before is the only one that can be 
 drawn from A. 
 
 Theorem : Through ez^ery point one^ and only one, 
 line can he drawn perpendicular to any given line. 
 
 § 107. We have thus obtained three special angles, 
 which have a definite magnitude at whatever point 
 in the plane we take the vertex. These are, the angle 
 of 7'otation^ the angle of continuation., and the r/^>^/ angle. 
 
 An angle of continuation is equal to two, and 
 an angle of rotation to four, right angles. 
 
 Any two angles which are together equal to a right 
 angle are said to be complementary., and either is called 
 the complement of the other. Two angles which are 
 together equal to an angle of continuation, or to two 
 right angles, are said to be supplejn^vifary, and either is 
 called the supplement of the other. 
 
 If two angles are together equal to an angle of 
 
Complementary Angles. 6i 
 
 rotation, or equal to four right angles, the one is some- 
 times called the completion of the other. 
 
 If we denote angles by small Greek letters, and an 
 
 angle of continuation by tt, a right angle becomes _ 
 
 2 
 
 and an angle of rotation 27r; consequently the above 
 statements may be expressed as follows : 
 
 two angles a and /3 are complementary if a 4-/3=-, 
 
 2 
 
 they are supplementary if a + /3=7r, 
 
 and one is the comphtion of the other if a-f /3=2r. 
 
 From these definitions it follows that 
 
 Theorem : Equal angles have equal complements^ 
 equal supplements^ ajid equal co7npletiom) and conversely 
 
 Angles which have equal complements^ or suppleme7its , 
 or completions^ are equal. 
 
 If two angles of the same sense are supplementar}', 
 each will be less than an angle of continuation or less 
 than two right angles. Hence they may be both right 
 angles, or else the one will be greater, the other less, 
 than a right angle. 
 
 An angle is said to be acute if it is less in magnitude 
 than a right angle, and obtuse if it is greater in magni- 
 tude than one a?td less than tzvo right angles. 
 
 § io8. If the non-coincident limits of two adjacent 
 angles are one the continuation of the other, then the 
 two angles are supplementary, as follows immediately 
 from the definitions. 
 
 The converse also holds, viz. 
 
 Theorem : If tiuo adjacent angles are supplement- 
 aty, the non-coincident limits are one the continuation 
 
Fig. 
 
 62 Elements of Plane Gecmetry. 
 
 of the other ^ or they are in a line. For the sum of the 
 two angles equals an angle of continuation. 
 
 § 109. If an angle ab \^ turned about the vertex 
 without changing its magnitude, then 
 both limits describe equal angles. For 
 if a and b (fig. 35) are turned to 
 a' and b\ so that ab-=a!b'^ then 
 ab-\-ba' -^ba' -^-a'b' \\i^vi<z^aa' =-bb' . 
 In the same manner it follows 
 Theorem : If a pencil of lines be 
 turned about its ce7itre without chang- 
 ing the angles betiaeen its rays, then 
 "■ all lines describe equal angles. 
 
 We may, therefore, speak of the angle through 
 which a figure has been tur7ied in its plane about a 
 point, if we understand hereby the angle described by 
 any ray drawn through that point. 
 
 If, in particular, an angle ^t ^ is turned through 
 an angle of continuation, it will in the new position 
 be vertically opposite to the original angle. Hence 
 
 Theorem : Vertically oppmte angles a?'e equal in 
 7?iagnitude and sense ; or 
 
 The two halves of a ray which turns about a point 
 in it describe angles which are equal in magnitude and 
 sense. 
 
 § no. The last proposition includes 
 The bisectors of vertically opposite angles arc in a 
 line. 
 
 Vertically opposite angles are symmetrical with 
 regard to their common vertex as centre of symmetry. 
 
 Two lines of which the sense has not been fixed 
 determine two i)airs of vertically opposite angles. 
 
Exercises. 63 
 
 Hence the bisectors of these four angles lie in two 
 lines. Of the four given angles two adjacent ones are 
 supplementary ; their bisectors include one-half of 
 each angle, and therefore half an angle of continua- 
 tion ; or 
 
 Theorem : The bisectors of the angles formed by 
 two intersecting lines are two lines which are at right 
 angles to one another. 
 
 Exercises. 
 
 (1) \i A andi? are two points in a line, determined by their 
 distance OA and OB from a fixed origin 0, and if M is the 
 mid point of A B, prove that 
 
 OM ^ ^{OA + OB). 
 
 State this result in words. 
 
 (2) HA, B and C, D are fom- points in a line, il/and -/V the 
 mid points o[ AB and CD respectively, then 
 
 AfJV= i [AC+BB) = ^{AD + BC). 
 
 State this result in words. 
 
 (3) If from the ends of a segment A B segments A A' and 
 BB' are set off which are equal in length but of opposite sense, 
 so that A A' = B'B, prove that the mid point of AB is the 
 same as the mid point of A'B'. 
 
 (4) If A', B', C denote the mid points respectively of the 
 three segments BC, CA, AB, determined by any three points 
 A, B, C in a row, and if denotes any other point in the row, 
 prove that 
 
 OA + OB+OC= OA'+OB'+OC. 
 
 (5) HA, B, C, and are any four points in a row, if ^', B' 
 denote the mid points respectively of the segments A B and B C, 
 and yJ/the mid point of A' B\ prove that 
 
 2{0A'+0B')= OA + 2OB+OC. 
 
 (6) If .^, B, C, D, and O are points in a row, A', B', C the 
 mid points respectively of the segments A B, B C, CD, if 
 
64 Elements of Plane Geometry. 
 
 further, A\ B'' are the mid points oi A' B' and B' C\ and if M 
 is the mid point of A" B", prove that 
 
 SOAI= OA + ^OB+sOC+OD. 
 
 (7) If « and ^ are two rays in a pencil determined by the 
 angles they make with an initial line <?, and if ;;/ is the bisector 
 of the angle a d, prove that 
 
 om = ^{oa + od). 
 
 State this result in words, and show that it is reciprocal to that 
 contained in exercise (i). 
 
 (8) State and prove the theorem reciprocal to that contained 
 in exercise (2). 
 
 (9) State and prove the theorems reciprocal to those con- 
 tained in exercises (3), (4), {5), and (6), 
 
 (10) From the vertex C of a given angle a // two straight 
 lines w and / are drawn, dividing that angle respectively into 
 two equal and into two unequal parts. Prove that half the 
 difference between these unequal parts is equal to the angle /n/>, 
 all angles being measured in the same sense. 
 
 (11) If the line / in the last question were not drawn between 
 the limits of the given angle, would the proposition in the last 
 exercise still hold ? 
 
 Show the connection between the propositions in exercises 
 (7) and (10). 
 
 (12) From a point four half-rays a, />, c, </are drawn. The 
 angles ad and cd are bisected by the half- rays w and ;/. Prove 
 that the angle ?nu is equal to half the sum of the angles ac and 
 dd, account being taken of the sense. 
 
 Show that this is essentially the same as exercise (8). 
 
 (13) Four straight lines are drawn from a point so that of 
 the four angles they form every two which are not adjacent are 
 equal to one another. Prove that the four constitute, in reality, 
 only two straight lines. 
 
 (14) Two angles, each less than an angle of continuation, are 
 supplementary, and the greater is double of the less. Find 
 what fraction the less is of a rigiit angle. 
 
Parallel Lines, 65 
 
 DRAWING. 
 
 To bisect a segment AB, mark a point A' on it which ap- 
 pears to you to be about the mid point, and set off from B to- 
 wards ^ a segment i9j5' equal in length to A A'. Then the 
 mid point oi AB will be the same as that of the smaller seg- 
 ment A' B', and may be guessed at much more accurately. 
 Assume it to be at A", and repeat the above process with A" 
 instead of ^' by setting off a segment B B" equal in length to 
 A A". Repeat the process if necessary. The second or third 
 trial will, as a rule, be sufficient. 
 
 Later on we shall see how to find the mid point, not by re- 
 peated trials, but by a geometj'ical conslriiclioji, which would 
 give the mid point at once with absolute accuracy, if the lines 
 which we draw were without thickness. For practical drawing 
 the method described is the quickest and the most accurate one. 
 
 (15) Bisect a segment by the method described. . 
 
 (16) Draw the figures to the exercises (i) to (6). 
 
 ( 1 7) Draw any triangle ABC, that is a figure formed by three 
 points A, B, C, called the vertices, and by the segments B\C, 
 CA, and A B, called the sides of the triangle. Bisect the sides 
 and join their mid points to the opposite vertices, viz. the mid 
 point of v? C to A, and so on. 
 
 If your drawing is accurate these lines will meet in a point. 
 
 CHAPTER V. 
 
 PARALLEL LINES. 
 
 § 1 1 1. Two indefinite lines in a plane may intersect, 
 as we have seen. We shall now consider the pos- 
 sibility of there being such lines which do not in- 
 tersect. 
 
 In order to do this let us take, in a plane, a line a 
 and a point F without it, and let us draw through 
 
66 Elements of Plane Geometry. 
 
 P a half-ray h which terminates at P. Let this half- 
 ray cut the line a in A. If we now suppose this 
 half-ray b to turn about P in the sense indicated 
 
 by the arrow, the point A 
 where it cuts the line a 
 will move to the right along 
 a, and will successively as- 
 sume the positions^, C, . ., 
 If we continue turning, this 
 point will move to a greater 
 b and greater distance from 
 
 its original position A, till at last it disappears to the 
 right in the indefinite production of the line a. 
 
 If we next turn the half-ray b from its original 
 position PA in the opposite sense, its intersection with 
 a will move to the left, assuming successively the posi- 
 tions D,E . . ., till it again disappears, but now to 
 the left. 
 
 As soon as the point of intersection has disappeared 
 to the right, the half-ray b, which terminates at P, does 
 not cut the line a any longer. But perhaps its produc- 
 tion beyond P does ? Certainly if we turn far enough 
 the production of b will meet a again. 
 
 § 1 1 2. Let b' be the position of the half- ray b at 
 the moment when the point of intersection with a 
 disappears to the right, and similarly let b" be the 
 position of b at the moment when the point of inter- 
 section disappears to the left, and let us now produce 
 both half- rays beyond P. We have then, instead of 
 two half-rays terminated at F, two indefinite lines 
 through P, and these will either coincide or include 
 an angle. 
 
Parallel Lines. 6j 
 
 If they include an angle, the production of b' will 
 fall either between a and b'', and that of b" between a 
 and b'^ as in fig. 37,01 these Fig. 37. 
 
 productions will lie on the ,r ^. 
 
 opposite sides of b" and b' '~^^^^Z^::^^^'"^^--~---~~~ 
 
 respectively, as in fig. 2>^. a — — — 
 
 If they coincide, the pro- 
 duction o^ b' will coincide with b", as in fig. 39. 
 
 § 113. These three are the 
 only cases conceivable. The 
 first contradicts our axiom "- ' ~~'^~" — V 
 
 of the line, as we shall pre- ^^ 
 
 sently see, and is therefore inadmissible. But our 
 axioms are not sufficient to ^ 
 
 Fig. 39. 
 
 decide which of the remain- -p g, 
 
 ing two cases actually does ' 
 
 occur. In looking at the ''' 
 
 figures the reader will at once feel that the third case 
 is the true one. But this cannot be considered decisive ; 
 for the two fines may include a very small angle — that 
 is, they may very nearly coincide without actually doing 
 so. Or it may be that sometimes the one, sometimes 
 the other, happens, according as we take the point P 
 at a smaller or greater distance from the line a. 
 
 The only way of settling this point is to make an 
 assumption, and to see whether the consequences 
 drawn from it do or do not agree with our experience. 
 
 The assumption to be made is, that the third case 
 only happens, and this will give us a new axiom. 
 
 Before, however, stating this axiom as a distinct pro- 
 position, let us see what the three cases really mean. 
 
 § 114. In the first case (fig. 37) every line through 
 
6S Elements of Plane Geometry. 
 
 P will cut the line a ; for if whilst turning round P 
 the point of intersection disappears to the right, the 
 production of the half-ray b will cut the line a to the 
 left. That is, through the point P no line can be drawn 
 which does not meet a. But every line within the 
 angle formed by b' and the continuation of b'' w^ould 
 cut a at both sides, hence at two points at finite dis- 
 tances. This contradicts our fourth axiom (§45), and 
 is therefore excluded. 
 
 In the second case (fig. 38) we should have to 
 turn b from b' through a finite angle before the con- 
 tinuation would cut a again, or there would be an 
 indefinite number of lines through P which do not 
 cut a. But in the third case (fig. 39) there w^ould be 
 only one line b' or b'' through P which does not cut a. 
 As soon as we turn this line about P it would meet a 
 either to the right or to the left. 
 
 § 115. Thus w^e are led to the conclusion that 
 there exist lines in a plane which, though both be un- 
 limited, do not meet. Such lines are called parallel 
 lines, ox pai'allels. 
 
 Definition of Parallel Lines. — Two nnln/iifed 
 lines in a plane wliich do not meet are called parallel 
 lines. 
 
 § 116. The assumption mentioned in § 113 may 
 now be stated thus : — • 
 
 Axiom VI. Through a given point only one line 
 can be drawn parallel to a giv en line. 
 
 § 117. This will hold for any line and any point 
 in space, for we can always draw a plane through a 
 point and a line. 
 
 It is further to be noticed that the statement 
 
Axiom about Parallels. 69 
 
 that two lines are parallel always includes the two 
 
 statements — ist, i/ie two lines lie in a plane ; and 2nd, 
 
 they do not meet. 
 
 Two lines in space as a rule do not meet, as we 
 
 have seen before. Two lines in a plane, however, 
 
 do meet, unless they are parallel, and this is an 
 
 exceptional case ; for through a point we can draw 
 
 an unlimited number of lines which meet a given line, 
 
 and only one which is parallel to it. 
 
 § 118. Let us next suppose that in a plane two 
 
 unlimited lines a and h are given (fig. 40), and let 
 
 us cut them in two distinct points by a third line t. 
 
 Such a line by which a given figure is cut is called a 
 
 transversal. At each of the 
 
 points where the transversal 
 
 cuts the given lines four angles 
 
 are determined, o, /3, y, with a. 
 
 and a', /3', 7', I' with b. We 
 
 shall consider these angles with- 
 out for the moment taking 
 account of their sense. 
 
 Four of these angles are between the given lines 
 (viz. y, ^, /3', and a'), and are called interior ajigles ; 
 the other four (viz. a, /3, y', V) lie outside a and h, 
 and are called exterior angles. Four lie on each side 
 of the transversal. Angles, one at each point, which 
 lie on the same side of the transversal, the one exterior, 
 the other interior, like o and a', c and h', &c., are called 
 con-esponding angles. 
 
 Tw^o angles on opposite sides of the transversal, 
 and both interior or both exterior, like a and y\ c and 
 jS', &c., are called alternate angles. 
 
70 Elements of Plane Geometry. 
 
 § 119. Among these eight angles there are, at each 
 of the points, two paks of vertically opposite, and there- 
 fore equal, angles. If it happens that two correspond- 
 ing angles are equal, for instance o=a', then their 
 supplements will be equal ; hence /3=/3'. As a=y, 
 /3=?, a'=y', /3'=3', these being vertically opposite 
 angles, it follows that a=y=a'=y' and/3=3=/3'=c', 
 and likewise that interior or exterior angles on the 
 same side of the transversal are supplementary. For 
 instance, y and /3' or a and 0' are supplementary. 
 The same is true if two alternate angles are equal, 
 for instance a=y', or if two interior or two exterior 
 angles on the same side of the transversal are supple- 
 mentary. For if I and a' are supplementary, then 
 a=a', as they have the same supplement I. 
 
 Hence if two coiresponding or two alternate angles 
 are equal, or if two interior or two exterior angles on 
 the same side of the transversal are supplementary^ 
 then every angle is equal to its corresponding and 
 to its alternate angle, and is supplementary to the 
 angle on the same side of the transversal which is 
 interior or exterior according as the first is i?itei'ior or 
 exterior. 
 
 § 120. Let us suppose that one, and therefore 
 all, of these conditions are satisfied ; then the angles 
 on the one side of the transversal are equal to those 
 on the other, and the figures on both sides are con- 
 gruent. We may, in fact, take the figure to the right 
 of the transversal, turn it round in the plane about 
 the mid point Af o(AB (fig. 41) till A flills on 
 B and B on A; then A£ will fall on BH, as 
 the angles EAB and HB A are equal, and B F 
 
Parallel Lines. yi 
 
 will fall on A G. Hence the figures coincide. If, 
 
 therefore, the lines A G and 
 
 BH intersect when produced, ^ 
 
 the lines AE and BF would ,/ 
 
 do so too, and the lines GE, / 
 
 HE would have two points /m 
 
 in common, which is impos- ^ L ^ 
 
 sible, according to Axiom IV. / 
 
 (§ 45). Consequently the two ° 
 
 lines do not intersect, that is, they are parallel. 
 
 This gives the following 
 
 Theorem : If two lines^ ad by a transversal^ 
 make corresponding or alternate angles equal, or interior 
 or exterior angles on the same side of the transversal 
 supplementary, the lines are parallel. 
 
 It follows, for instance, that 
 
 ' ' Fig. 
 
 lines which are perpendicular to 
 the same line are parallel. 
 
 § 121. The last theorem shows 
 how we may proceed if we have to 
 draw through a given point A a line 
 parallel to a given line a. 
 
 We draw through A a line AB 
 cutting a at B and making some angle, say a, with it; 
 we then move this angle a by sliding the one limit 
 along A B until its vertex comes to A. Its second limit 
 will then be parallel to a, because it makes alternate 
 angles equal. This process is practically carried out 
 on the dra\\-ing board by aid of set squares. 
 
 § 122. Theorem : If two lines are parallel, then 
 r^ery transversal makes cor?'esponding angles equal, 
 alternate angles equals and interior or exterior angles on 
 
'7-2 Elements of Plane Geometry. 
 
 the same side of the transversal suppleme^itary. For 
 there is only one parallel, and this may be found by 
 making alternate angles equal. 
 
 § 123. This may also be stated thus : 
 
 Theorem : The necessary and sufficient condition 
 that tivo lines in a plane may be parallel is that a 
 transversal makes corresponding angles equal. And 
 
 If co7'responding angles are not equals the two lines 
 are not parallel ; that is, they will meet if sufficiently 
 produced. 
 
 It follows also that any two parallels have the 
 property proved in § 120 for lines making correspond- 
 ing angles equal, viz. that if we take the mid point 
 J/ of any transversal AB (fig. 41), and turn the whole 
 figure through an angle of continuation about M, the 
 two halves of the figure on the two sides of the trans- 
 versal will have interchanged places. This may be 
 expressed, as in § 98, by saying that the figure is 
 symmetrical with regard to J/ ; or 
 
 Any two parallels ai'e symmetrical with regard to 
 the mid point of the segment which they determine on 
 any transversal. 
 
 § 1 24. Pencil of Parallels. — We may have in a 
 plane more than one line parallel to a given line. 
 For all such lines the following propositions hold : 
 
 Theorem : If two lines in a plane be parallel to a 
 third, they are parallel to each other. For if the two 
 lines were not parallel they would intersect in a point, 
 and through this point two lines would pass, both 
 parallel to the third ; but this is against our axiom. 
 
 Further, a series of parallel lines is cut by any other 
 line under equal angles, or the line is equally inclined 
 to them. 
 
ScVisc in Parallel Lines. 73 
 
 The aggregate of all lines in a plane which are 
 parallel to one line, and therefore parallel to each 
 other, is called 2i pencil of parallels. 
 
 § 125. Sense in Parallels.— If AB and A' B' 
 be two segments in two parallel Fig. 43. 
 
 lines, we say these are of the 
 same sense if both He on the same 
 side of the line A A' (fig. 43), 
 but they are of opposite ^ sense 
 if they He on opposite sides of 
 the line A A' (fig. 44). / 
 
 This enables us to fix the sense 
 of one line by reference to that of a parallel line. 
 
 We need only take in each of ^"^^- '•'<' 
 
 the parallel lines a segment of the 
 same sense as the line. The 
 parallels are then said to be of 
 the same or of opposite sense, 
 according as the two segments are 
 of the same or of opposite sense. 
 
 Taking account of the sense, we may now complete 
 the theorems about angles between parallels cut by 
 a transversal as follows (fig. 45) : 
 
 Corresponding or alternate 
 angles are equal in magnitude 
 a7id sense if the pa7'allels have 
 the same sense ; and, conversely, 
 if corresponding angles are equal 
 in magnitude and sense the lines 
 are parallel and of the same sense. 
 
 § 126. If we cut two parallels a and a' by two other 
 parallels b and b' , and we take one of the four angles 
 
74 Elements of Plane Geometry. 
 
 between ah and one of the angles between a'h^ then 
 these two angles are either equal or supplementary. 
 For we may consider /^ as a transversal by which the 
 two parallels a and a' have been cut. If we take in 
 the same way one of the angles a'b and one of the 
 angles a' b\ then these are either equal or supplement- 
 ary. Hence also any one of the angles ab, and any 
 one of the angles a' b', are either equal or supplement- 
 ary. This may be stated thus : 
 
 If two angles in the same plane have the limits of the 
 one parallel to those of the other, the tiuo angles are either 
 equator supplementary. 
 
 If we take account of the sense of the lines and 
 angles, we get at once the following 
 
 Theorem: Two angles which have their li^nits 
 parallel are equal if the limits of the one are both of 
 the same or both of the opposite sense to the limits of 
 the other; but the two angles are supplementary if one 
 of the limits of the one angle is of the same, and the other 
 of the opposite, sense to the parallel limit of tlie other 
 angle. 
 
 Exercises. 
 
 (i) If two lines be each perpendicular to a third, they will 
 be parallel to one another. 
 
 (2) vState and prove the converse to the above. 
 
 (3) Two lines of Avhich one is not, whilst the other is, perpen- 
 dicular to a third line will necessarily intersect. 
 
 What logical connection has this proposition with the two 
 preceding ones ? 
 
 (4) No two lines which are perpendicular respectively to two 
 intersecting lines can be parallel to one another. 
 
 Find the logical connection between this and the above. 
 
 (5) If the limits of an angle are perpendicular to those of 
 
Exercises. 75 
 
 another, the angles will be either equal or supplementary, 
 according as they are of the same or of opposite sense. 
 
 (6) State the converse of exercise (5). Is it true? 
 
 State and prove the converse of (5) after having changed it to: 
 Of two angles of like sense the first limit of one is perpendi- 
 cular to the first limit of the other ; then if the second limit of 
 the one is perpendicular to the second of the other the angles 
 are equal. (Convert only the statement following the ' if '.) 
 
 (7) If the limits of one angle be perpendicular to those of 
 another, then the bisectors of these angles will either be perpen- 
 dicular or parallel. 
 
 (8) Investigate the converse theorem, as in exercise (6). 
 
 DRA WING. 
 
 A set square, which we now add 10 our drawing instruments, 
 is commonly made of wood or ebonite in the form of a triangle 
 which has one right angle and the other two either equal or 
 one of them half the size of the other. The student should 
 get one of each kind. 
 
 To draw parallel lines we place a set square with one of 
 its edges against a straight-edge and slide it along the latter. 
 The different positions of either of the other edges will give 
 parallel lines (§ 121), of which we may draw as many as we like. 
 
 To draw perpendiculars the right angle of a set square is 
 made use of. If the hypothenuse is made to slide along a 
 straight-edge, and if in one of its positions a line be drawn along 
 one of the limits of the right angle, and in another position a 
 line along the other limit, then these lines will be pei-pendicular 
 (§ 126). 
 
 (9) Through a given point draw a line parallel to a given 
 line. 
 
 (10) Through a given point draw a line perpendicular to a 
 given line, both in the case where the given point is, and where 
 it is not, on the given line. 
 
 (11) Draw any triangle, and through its vertices dr»w pa- 
 rallels to the opposite sides. These will form another triangle. 
 
*j6 Elements of Plane Geometry. 
 
 (If your drawing is accurate the vertices of the first will be 
 the mid points of the vertices of the second triangle.) 
 
 (12) Draw any triangle, and through its vertices draw per- 
 pendiculars to the opposite sides. (If your drawing is accurate 
 these will meet in one point.) 
 
 (13) Draw an angle equal to the angle between two lines 
 which meet off your drawing paper. 
 
 (14) On aline set off a number of points A, B, C . . . "iX 
 equal distances. Through these points draw a series of parallel 
 lines a, b, c . . ., and in any other direction a second series of 
 parallel lines a', b', c' . . . Your paper will then be covered 
 with a net of lines. 
 
 If your drawing is accurate the intersections a'b, b'c, c'd . . . 
 will all lie in a line. So Avill the points a'c, b'd . . . Draw 
 these lines and all others of a similar kind which you can dis- 
 cover. 
 
 CHAPTER VI. 
 
 ANGLES IN POLYGONS. 
 
 § 127. Broken Lines. — If any number of points 
 A, B, C,D.. .he given in a plane, and these be joined 
 in any order, the first to the second, the second to the 
 third, and so on, by segments of lines which terminate 
 at these points, we obtain what is called a dro/cm line. 
 The first and last of the given points are its end poifits. 
 
 This broken line may be supposed described by 
 a point moving from A to B along their join, then 
 from B to C, and so on. 
 
 The segments AB, BC, CD . . . are called the 
 sides of the broken line. Each side, as well as the 
 whole broken line, has a sense. 
 
Angles of a Broken Line 
 
 77 
 
 Fig. 46. 
 
 At each of the given points, with the exception of 
 the end points, two sides meet, one having there its 
 end point and the other its origin. 
 
 These points are called the rertices of the broken 
 line. 
 
 § 1 28. Tlie nuinher of sides is one less than the mem- 
 ber of given points J for every 
 one of the points, excepting 
 the last, is the origin of one side. 
 
 The nmnher of vertices is two 
 less than the number of given 
 points, for every one, with the 
 exception of the end points, is 
 a vertex. 
 
 § 129. Angles of a Broken Line. — At each of the 
 vertices two sides meet and determine two angles, of 
 which one is less, the other greater, than an angle of 
 continuation. 
 
 If we turn the first side A B about ^ in a given 
 sense till it coincides with the next side B C, it describes 
 one of the angles at B. If we next turn B C about C 
 in the same sense to CD, then CD about D io D E, 
 and so on, always turning 
 in the same sense, we get 
 at each vertex one angle. 
 Every side is, then, the 
 limit of two angles which 
 lie on the same side of it. 
 
 The angles thus deter- 
 mined are called the angles 
 of the broken line taken in the given sense. 
 
 If we change the sense of turning, every angle 
 changes into its completion. 
 
78 
 
 Elements of Plane Geometry 
 
 Fig. 48. 
 
 § 130. It may happen that in one of these two sets 
 of angles every one is less than an angle of continua- 
 tion. In this case the broken line has the following 
 property : if one side be produced both ways, then the 
 two adjacent sides lie on the 
 same side of it. The broken 
 line is then said to be convex^ 
 and in this case those angles 
 only are called the angles of 
 the broken line which are each 
 less than an angle of continua- 
 tion. 
 
 Thus the figures 46 and 47 show convex broken 
 lines, whilst the broken line in fig. 48 is not convex, 
 for the sides DE and FG are on opposite sides of 
 the side EF. 
 
 The broken line in fig. 49 also is not convex. 
 § 131. Exterior Angles. — If in a broken line 
 every side be produced in 
 the sense in which a moving 
 point describes the broken line, 
 we obtain at each vertex an 
 angle between the one side 
 produced and the next follow- 
 ing side. This angle is called 
 an exterior angle. 
 
 We thus have at every ver- 
 tex an exterior angle. These are all to be taken in 
 the same sense. 
 
 § 132. Polygons. — If in a broken line the two end 
 points coincide, the figure obtained is called 2i polygon 
 (figs. 50 and 53), and the broken line \\.s peri meter. 
 
Polygons. 79 
 
 A polygon has as many vertices and angles as it 
 has sides. The angles are determined as in the case 
 of the broken line. 
 
 A polygon has a sense. According as the peri- 
 meter is described in the one or in the opposite 
 sense, we say that the polygon itself is of the one or 
 the opposite sense. 
 
 § 133. If no side of a polygon cuts another it 
 bounds one finite connected part of the plane, which 
 is called the area of the polygon, and we may speak 
 of points within and without the polygon. 
 
 In this case we understand by the angles of the 
 polygon those angles of which the part near the vertex 
 lies within the polygon, these angles being all taken 
 in the same sense. If each of these angles is less 
 than an angle of continuation, the polygon is said to 
 be convex. If one or more angles are greater than 
 an angle of continuation, the polygon is said to have 
 one or more re-entrant angles, provided that no two 
 sides of the polygon intersect. 
 
 § 134. If the polygon cuts itself once or several 
 times, then not a single part of Fig. 50. 
 
 the plane is bounded by it, but 
 two or more disconnected por- 
 tions. 
 
 In this case it depends, as 
 in the case of the broken line, 
 "Upon the sense in which the 
 angles are taken, which of the two angles between two 
 consecutive sides we take as the angle of the poly- 
 gon, Thms in the polygons (figs. 50 and 51) we may 
 take -either the angles shaded in the first, or those 
 
8o 
 
 Eleviciits of Plane Geometry 
 
 Fig. 
 
 shaded in the second, as the angles in the polygon ; 
 but it would be against our defini- 
 tion to consider those shaded in 
 fig. 52 as the angles in the po- 
 lygon, for the side B C or DA 
 would have the two angles adja- 
 cent to it on opposite sides of 
 it. 
 A polygon has at least three sides. In 
 this case it is called a triangle^ 
 and is always convex. 
 
 Polygons with 4, 5, 6, 8, 10, 12 
 
 sides are respectively called tet?-a- 
 
 gon or quadrilateral, pentagon, 
 
 hexagon^ octagon, decagon, duode. 
 
 cagon. 
 
 A polygon which has all its angles equal, and also 
 all its sides equal, is said to be regular. A regular 
 polygon is either convex or its sides cut each other, 
 but it cannot have re-entrant angles. A regular poly- 
 gon whose sides cut each other is also called a star 
 polygon. 
 
 § 136. Diagonals. — Any 
 line joining two vertices is a 
 side if the two vertices are 
 consecutive ones, otherwise it 
 is called a diagonal. Thus 
 in fig. 53 the lines AC, AD, 
 ^c.y are diagonals; whilst in 
 fig. 50 the line ^C or BD 
 is a diagonal. 
 
 § 137. Let us consider the simplest of all polygons 
 
 „->^ 
 
Sum of A^igles in a Triangle, 8i 
 
 the triangle, and in a triangle ABC let us produce 
 (fig. 54) a side AB to D\ then the angle CBD 
 contained by the side CB and the side A B produced 
 will be equal to the sum of fig. 54. 
 
 the two angles at Cand A in ,C 
 
 the triangle. For if through B 
 aline BE he, drawn parallel to 
 A C, it will divide the angle 
 CBB) into two parts. Of ^^ 
 these angle BBB> is equal to CA B, they being corre- 
 sponding angles, and angle CBE is equal to A CB, 
 they being alternate angles. But the angles ABC, 
 CBE, and EB D form together an angle of continua- 
 tion, and as they are equal respectively to the three 
 angles in the triangle, we have 
 
 Theorem : The sum of the angles in any triangle 
 is equal to an angle of continuation, or equal to two 
 right angles, or =7r. 
 
 § 138. At the same time it has been proved 
 
 Theorem : In every triangle an exterior angle is 
 equal to the sum of the two interior and opposite 
 angles) or 
 
 Eveiy angle in a triangle is supplementary to the 
 sum of the other two. 
 
 This shows also that in a triangle at least two 
 angles are acute. The third angle may be acute, or 
 right, or obtuse ; and the triangle is called acute-angled, 
 right-angled, or obtuse-angled accordingly. 
 
 In a right-angled triangle the side opposite the 
 right angle is called the Hypothenuse. 
 
 § 139. These are very important theorems. They 
 are immediate consequences of the axiom about 
 
S2 Elements of Plane Geometry. 
 
 parallels. If that axiom be not the true one, we ought 
 to have made in § 113 the other possible assumption. 
 But it can be proved that the sum of the angles 
 in a triangle would in that case be less than two right 
 angles, by a quantity which increases with the size of 
 the triangle. 
 
 Angles, however, can be measured with very great 
 accuracy, and in many triangles the angles have been 
 measured. In every case, even with triangles having 
 sides many miles long, the sum of the angles has been 
 found to be equal to an angle of continuation, at 
 least so nearly that the difference may be accounted 
 for by inaccuracy in measurement. If we ought 
 to have made in § 113 the other assumption, this 
 sum should have been found different from an 
 angle of continuation. Experience thus confirms our 
 axiom. 
 
 § 140. Every convex polygon may be divided by 
 diagonals into triangles. The simplest way of doii.g 
 this is by drawing all diagonals 
 which pass through one vertex 
 A. If we leave out the two 
 sides adjacent to A, in the 
 figure the two sides AB and 
 A G, every other side of the 
 polygon determines one triangle 
 with a vertex at A. The num- 
 ber of these triangles is thus two less than the number of 
 sides. Hence a convex polygon of ti sides can be 
 divided into n-2 triangles, such that their angles 
 together make up the angles in the polygon. But the 
 sum of the angles in each triangle is equal to an angle 
 
Siiin of Angles in a Polygon. 83 
 
 Df continuation. Therefore the sum of all the angles 
 in the convex polygon is equal to n-2 angles of con- 
 tinuation, or equal to 2{n-2) right angles. Hence 
 
 Theorem : The sum of all the angles in a convex 
 f)olyg07i is equal to as many angles of continuation less 
 two as the polygon has sides ; or, 
 
 In a convex polygon of n sides the sum of all the 
 angles equals 2n-4 right angles. 
 
 Thus the sum of the angles in a convex quadri- 
 lateral is equal to four right angles. 
 
 § 141. Sum of Exterior Angles. — If all the sides 
 of a convex polygon be produced in the same sense, we 
 get an exterior angle at every vertex. This is supple- 
 mentary to the adjacent angle in the polygon. In 
 a convex polygon of n sides we have, therefore, 7t exte- 
 rior angles, which, together with the adjacent interior 
 angles, form n angles of continuation. But the interior 
 angles are together equal to 71-2 angles of continuation ; 
 therefore 
 
 Theorem : Li any coiivex polygon the su7n of the 
 exterioi' afigles (one at each vertex) equals two a7igles 
 of continuation^ or four 7'ight a7igles. 
 
 S 142. These two theorems about 
 
 ^ . Fig. 56. 
 
 the angles in a polygon have been 
 proved for convex polygons only. 
 The first is also true for polygons 
 with re-entrant angles, provided that 
 no side cuts another. The proof 
 is very similar. It is always pos- 
 sible to divide the polygon by 
 diagonals into n-2 triangles (fig. 56). The complete 
 proof will be left to the student. The second theorem 
 
84 Elements of Plane Geometry, 
 
 about the exterior angles, however, does not hold in 
 the above form for other than convex polygons. 
 
 Exercises. 
 
 (i) In every right-angled triangle the two acute angles are 
 complementaiy. 
 
 (2) If one angle of a triangle be equal to the sum of the other 
 two, what is its magnitude ? 
 
 (3) State and prove the converse to the proposition in exercise 
 (i), and show that the answer to exercise (2) is a logical conse- 
 quence of (i) combined with the theorem in § 137. 
 
 (4) Find the value of the sum of the angles in a convex 
 polygon of four, five, six, seven^ and eight sides. 
 
 (5) Find the value of an angle in a regular convex polygon 
 of three, four, six, eight, and twelve sides. 
 
 (6) Prove that a convex polygon cannot have more than three 
 obtuse exterior angles, and not more than three acute interior 
 angles. . 
 
 (7) The bisectors of two angles in a triangle, produced till 
 they meet, include an obtuse angle. 
 
 (8) If the sides of one triangle be respectively perpendicular 
 to those of another, the angles of the one will be respectively 
 equal to those of the other. 
 
 (9) Determine the magnitude of the acute angles in each of the 
 two set squares described in Drawing Exercises to Chapter V. 
 
 (10) Through a point on a line, and on the same side of the 
 line, four half-rays are drawn, one perpendicular to the given 
 line and the others making angles with it equal to the acute angles 
 of the set squares mentioned in the last exercise. Determine 
 the values of the angles contained by these rays. 
 
 DKA WING. 
 
 The first seven of the following problems should be solved 
 by aid of the set squares only. 
 
 (11) Draw the figures to exercises (8) and (10). 
 
 (12) Divide a right angle into two, into three, and into six 
 ci[ual parts. 
 
Symmetry. 85 
 
 (13) Draw angles which have respectively the magnitude of 
 i, A, §, |, and I of a right angle. 
 
 (14) Draw triangles which have their angles equal respectively 
 to the angles in the set squares. 
 
 {15) Draw a triangle which has all its angles equal. [See 
 exercise (5).] 
 
 (16) Draw a quadrilateral and also a hexagon which have all 
 their angles equal. 
 
 (17) Through a given point in a given line draw two half- rays- 
 so that the angle between them is bisected by the given line. 
 
 This may be done in eleven different ways, according as the 
 angle between the given line and one of the half-rays equals 
 I, 2, 3 . . . 10, or II times | of a right angle. 
 
 (18) Draw a convex polygon which has three of its interior 
 angles acute. 
 
 {19) Draw a polygon which has three of its exterior angles 
 obtuse. 
 
 CHAPTER VII. 
 
 AXIAL AND CENTRAL SYMMETRY. 
 
 - § 143. In the previous chapters the fundamental 
 notions of geometry and the elementary properties of 
 the simpler figures have been developed. In parti- 
 cular we have obtained a series of axioms, that is, of 
 propositions taken from experience. To these no new 
 ones will be added. Those already given are necessary 
 and sufficient to characterise space as we conceive it. 
 In what follows we shall develop further consequences 
 of these axioms, and of the propositions already derived 
 from them relating to segments and angles. 
 
 It ^\iU now be our task to investigate properties of 
 
86 Elements of Plane Geometry. 
 
 more complicated figures. These investigations will 
 be limited to properties of figures which are congruent 
 and to consequences flowing fi-om these properties. 
 
 § 144. Congruent figures have been defined (§ 39) 
 as figures which, when applied to one another, can be 
 made to coincide. In other words, tvvo figures are 
 congruent if it is possible to place the one on the 
 other in such a manner that every point in either falls 
 on, and coincides with, some point in the other. The 
 one figure is, in fact, an exact copy of the other. Both 
 have the same shape and the same size, but they 
 differ in position, and position only. 
 
 To a plane figure, a plane figure only can be 
 congruent. Their planes may be distinct, and may 
 lie anywhere in space, or else their planes may lie one 
 on the other. In the latter case, which is the one 
 with which we are most concerned, the two figures 
 occupy different positions in the same plane. But 
 even then we shall suppose that each figure lies in its 
 own distinct plane, that it is possible to separate these 
 and to move either figure with its plane into any po- 
 sition that may aj^pear convenient for our investiga- 
 tions. Of such positions two are of special interest. 
 
 § 145. Corresponding Points. — Let us suppose 
 that the two figures coincide. In this position every 
 point A in the one coincides with a point A' in the 
 other. These points will be said to correspond the 
 one to the other. Hence 
 
 To every point in one of two congruent figures, there 
 corresponds ofie, and only one, point in the other, those 
 points being called * corresponding^ ivhich coincide if the 
 two figures are applied to one another. 
 
Coincident Figures. 
 
 ^7 
 
 know that to any 
 
 Fig. 57- 
 
 Similarly we have to every line in the one a 
 corresponding line in the other, to ever)' segment or 
 angle in the one a corresponding segment or angle in 
 the other, and always to every part of the one figure 
 a corresponding part of the other. Hence 
 
 Definition : In two congruent figures those parts ore 
 catted corresponding which coincide if the whole figures 
 are made to coincide. 
 
 It follows that — Con-esponding parts of congruent 
 figures are themselves congruent. 
 
 § 146. This implies : if we know that two figures 
 are congruent, and if we further 
 two points A and j5 (fig. 57) in 
 the one there correspond the 
 two points A' and B' in the 
 other, then we know also that the .- 
 line through A' and B' corre- 
 sponds to the line through A and 
 B, and that the segments AB and 
 A'B' are equal in length. If, 
 besides, C and C are corre- 
 sponding points, then we know 
 that to the angle ABC there 
 corresponds the angle A'B'C, 
 and that these angles are equal ; 
 for if we apply the one figure ^'-^ 
 
 to the other these corresponding 
 points, segments, and angles Avill coincide. 
 
 In the same manner it will be seen that to the mid 
 point M of the segment AB corresponds the mid 
 point M' of the corresponding segment A'B'. 
 
 If we draw in the one figure from A a perpendicular 
 
 \ 
 
8S Elenie7tts of Plane Geometry. 
 
 to the line BC, and in the other figure from the 
 corresponding point A' a perpendicular to the line 
 B' C corresponding to B C, then these perpendiculars 
 will be corresponding lines. For if we apply the figures 
 again, A' falls on A, B' C on B C, and the perpen- 
 diculars must coincide, as only one perpendicular can 
 be drawn from a point to a line. 
 
 Thus we arrive at propositions like the following : 
 
 To the Join of tivo points {or lines) in the one figure 
 corresponds the join of the two coi'responding points {or 
 lines) in the other. 
 
 The distance between tive points in the one figure 
 is equal to the distafice between the two corresponding 
 points in the other. 
 
 The angle between two lines in the one is equal to the 
 angle betiveen the two corresponding lines in the other. 
 
 § 147. Symmetry with regard to an Axis. — All 
 these properties are true for every position of the two 
 figures. But if we wish to see which points and lines 
 are corresponding we must apply the one figure to the 
 other. This operation, however, can only be per- 
 formed as a mental conception. We must therefore 
 find other means for comparing the two figures. The 
 investigation of the properties required for such 
 comparison is greatly simplified by placing the two 
 figures in convenient positions, and we shall see 
 that two positions are particularly useful for this pur- 
 pose. 
 
 To obtain these, let us start with the position of 
 coincidence, and let us take in the common plane 
 any line s (fig. 58). We may then turn the plane of 
 the one figure about this line s till its ])lane, after half 
 
Axis of SyniDictry. 89 
 
 a revolution, coincides again ^^ith the plane of the 
 other figure. The two figures themselves will then have 
 distinct positions in the same plane. But they will 
 have this property, that they can be made to coincide 
 by tummg the one figure about the Hne j-, that is, by 
 folding the plane over along that line. 
 
 Two figures in the same plane which have this 
 property are said to be symmet7'ical with regard to the 
 line s as axis of sy 711111 etry. 
 
 An example of such symmetry we have had in 
 § 99, ^vhere it was shown that two intersecting lines 
 are symmetrical with regard to a bisector of their 
 angle. As a special case of this, or as a consequence 
 of the reasoning used in § 105, we see that a perpen- 
 dicular to a line a is symmetrical to its continuation 
 with regard to the Hne a as axis. In fig. 34 (§ 106) 
 the two points A and A^ are symmetrical ^^^th regard to 
 the line a as axis. 
 
 § 148. Symmetry with regard to a Centre. — 
 If, on the other hand, we take in the common plane 
 of the two coincident figures any point 6", instead of 
 a line s, we may turn the one figure about this point 
 so that its plane slides, whilst turning, over the other 
 plane without ever separating from it. 
 
 Let this turning be continued till one line through 
 S^ and therefore (§ 109) the whole figure, has been 
 turned through an angle of continuation about 5. 
 
 The two congruent figures still lie in the same 
 plane, and have such positions that the one can be 
 made to coincide with the other by turning it in the 
 plane through an angle of continuation about the 
 fixed point S (fig. 59). 
 
90 Eleinents of Plane Geometry. 
 
 Two figures which have this property are said to 
 be symmetrical with regard to the poi?tt S as centre of 
 symmetry. 
 
 Examples of central symmetry have already 
 occurred. Two points are symmetrical with regard to 
 the mid point of the segment joining them (§ 98). 
 Two vertically opposite angles are symmetrical with 
 regard to their common vertex (§ no), and from the 
 reasoning used in § 120 it follows that two parallel 
 lines are symmetrical with regard to the mid point of 
 any segment having one end-point in each. 
 
 § 149. We have thus obtained two kinds of 
 symmetry. The one is symmetry with regard to an 
 axis, called, for shortness, axial symmetry. The other 
 is symmetry with regard to a centre, or central sym- 
 metry. 
 
 These two kinds of symmetry stand in the relation 
 of reciprocity which has been explained in § 62. For 
 where we took a line in the one case we took a point 
 in the other, about which the planes containing the 
 two figures were turned. 
 
 It is therefore to be expected that both have a 
 great many reciprocal properties. These will be 
 brought prominently forward in §§ 1 51-153. 
 
 § 150. Two figures which are symmetrical are by 
 definition congruent. But it is also true that a?iy tivo 
 congruent plane figures can always be placed^ and this 
 in an infinite number of different ways, in such 
 positions that they are symmetrical with regard to an 
 axis or with regard to a centre. To do this we need 
 only apply the one figure to the other so that they 
 coincide, then select in their common plane any line 
 
SyjHinetry. 91 
 
 as axis, or any point as centre, and turn the one plane 
 about this line or point, as before described. 
 
 The statement that two figures are symmetrical 
 implies, therefore, first, that they are congruent, and 
 seco7idly, that they lie in particular relative positions. 
 
 Some of the properties following from the first con- 
 dition have already been stated. Everything relating to 
 the correspondence of parts in congruent figures holds 
 for symmetrical figures. These properties will be 
 shortly repeated, and then the additional properties 
 peculiar to the position of symmetry ^^dll be stu- 
 died. 
 
 The student should not omit to go through all the 
 exercises in drawing at the end of this chapter ; for 
 the special cases given there, and more particularly 
 the actual dra^ving of the figures, will greatly facilitate 
 the understanding of the general propositions laid 
 do\Mi in the next paragraphs. 
 
 The latter should be read again when these exer- 
 cises have been gone through. 
 
 As the two kinds of symmetry are reciprocal, it 
 follows that to a property relating to lines in the one 
 case we must have a property relating to points in the 
 other, and vice versa (§ 62). And similarly for a 
 property in either relating to segments we get in the 
 other a property relating to angles. 
 
 It is customary to print reciprocal theorems and 
 reciprocal investigations on opposite halves of a page, 
 broken into two columns, and this will be done in the 
 following paragraphs. 
 
 § 151^. Axial Sym- § 151^. Central Sym- 
 
 metry. — Definition : If metry. — Definition : If 
 
92 Elements of Plane Geometry 
 
 Fig. 
 
 Fifi. 59. 
 
 two figures in the same 
 plane can be made to coin- 
 eide by turning the one about 
 a fixed line in the pla?te 
 through a?i a?igle of conti- 
 nuation, the two figures are 
 said to be symmetrical with 
 regard to that line as 
 
 AXIS OF SYMMETRY. If 
 
 the two figures are halves 
 of one figure the whole 
 figU7'e is said to be sym- 
 metrical with regard to 
 the axis, and this axis is 
 said to be an axis of sym- 
 metry, or simply an axis 
 of the fig7ire. 
 
 § 152^. To a point, or 
 line, or angle, &'c., in the 
 one figure corresponds a 
 point, or line, or angle, &'c., 
 in the other figure. 
 
 ttvo figures in the same 
 plane can be made to coi?i- 
 cideby turningthe one about 
 a fixed point in that plane 
 tJu'ough an angle of con- 
 tinuation, the two figures 
 are said to be symmetrical 
 with regard to that point 
 
 as CENTRE of SYMMETRY. 
 
 If the two figures are halves 
 of one figure, the whole fi- 
 gure is said to be symme- 
 trical 7vith regard to a 
 centre, and this centre is 
 said to be a centre of 
 SYMMETRY, or simply a 
 centre of the figure. 
 
 § 152/^. To a line, or 
 point, or segment, i^c, in 
 the one figure corresponds 
 a line, or point, or segment, 
 i^c, in the other figure. 
 
General Theorems on Symmetry. 93 
 
 Fjg. 60. Fig. 6i. 
 
 / 
 
 To the join of two lines 
 corresponds the join of the 
 corresponding lines, to the 
 join of two points the join 
 of the corresponding points, 
 to the segment betn'cen tiuo 
 points the segment between 
 the corresponding points, to 
 the angle between two lines 
 the angle between the cor- 
 responding lines, and so on. 
 
 Fig. 62. 
 
 To the join of tzuo points 
 corresponds the join of the 
 corresponding points, to the 
 join of two lines the join of 
 the cor?'espondinglines, to the 
 angle betiveen two lines the 
 angle between the corre- 
 sponding lines, to the seg- 
 ment between two points the 
 segment between the corre- 
 spondifig points^ and so on. 
 
 Fig. 63. 
 
 \ 
 
 To three or more po hits 
 in a line correspond three 
 or mo7'e points in the cor- 
 responding line ; or 
 
 To three or more lines 
 in a point correspond three 
 or more lines in the cor- 
 responding point ; or 
 
94 
 
 Elements of Plane Geometry. 
 
 If three points A, B, C 
 (fig. 62) lie in a line, the 
 three corresponding points 
 A', B', C lie in a linewhich 
 corresponds to the first. 
 
 To three or more lines 
 in a point con-espond three 
 or more lines in the cor- 
 responding point \ or — If 
 three lines abc lie in a 
 point (fig. 64), the three 
 correspondi7ig lines dJ h' d 
 lie in a point which cor- 
 responds to the former. 
 
 Fig. 64. 
 
 If three lines a, b, c 
 (fig. d'^ meet in a point, 
 the three corresponding lines 
 a', b', c' meet in apoint which 
 corresponds to the former. 
 
 To three or more points 
 in a line correspond three 
 or more points in the cor- 
 responding line ; or — If 
 three points AB C(fig. 65) 
 lie in a line, the three 
 corresponding points A'B' 
 C lie in a line which cor- 
 responds to the forme?'. 
 
 Fig. 60. 
 
 Corresponding segments 
 are equal in length. 
 
 As they are not necessarily 
 parallel, we cannot compare 
 their sense. 
 
 Corresponding angles 
 are equal and of opposite 
 sense. 
 
 '^--^y^/ 
 
 Corresponding angles 
 are equal in mag?iitude and 
 sense. 
 
 Corresponding segments 
 are equal and of opposite 
 sense. 
 
General TJieorems on Symmetry. 
 
 95 
 
 To parallel lines corre- 
 spond parallel li7ies. 
 
 To a perpendicular cor- 
 responds a perpendicular. 
 
 To the bisector of an 
 angle or segment cor7'e- 
 sponds the bisector of the 
 corresponding angle or seg- 
 ment. 
 
 To equal segments or 
 angles correspond equal seg- 
 ments or aiigles. 
 
 Corresponding poly gons 
 are congruent but of oppo- 
 site sense (§ 132). 
 
 § i53<3;. Every point 
 in the axis corresponds to 
 
 To parallel lines cor- 
 respond parallel lines. 
 
 To a perpendicular cor- 
 responds a perpendicular. 
 
 To the bisector of a 
 segment or an angle cor- 
 responds the bisector of the 
 corresponding segment or 
 angle. 
 
 To equal angles or 
 segments correspond equal 
 angles or segments. 
 
 Corresponding polygons 
 are congruent and of like 
 sense{\ 132). 
 
 § 153^. Every line 
 through the centre cor- 
 
 itself 
 
 The axis 
 to itself 
 
 corresponds 
 
 1'esponds to itst 
 The centre 
 to itself. 
 
 corresponds 
 
 Fig. 66. 
 
 Fig. 67- 
 
 
 A. 
 
 a' 
 
 \ 
 
 r 
 
 % 
 
 4- 
 
 \/ 
 
 / 
 
 / 
 
 b/ 
 
 / 
 
 / 
 
 .9 ft 
 
 \4- 
 
 "a" / \ 
 
 
 
 \ 
 
 Every poi 
 corresponds to i 
 the axis. 
 
 nt which 
 tsclf lies in 
 
 Every li?ie which cor- 
 responds to itself lies in the 
 centre (passes through it). 
 
96 
 
 Elements of Plane Geometry. 
 
 The join of two cor- 
 responding lines lies on the 
 axis, 
 
 for it corresponds to itself. 
 
 Two corresponding lines 
 are equally inclined to the 
 axis ; their join lies on the 
 axis j or — 
 
 The angle between two 
 corresponding lines is tri- 
 sected by the axis. 
 
 The join of two cor- 
 responding points is pe7'- 
 pendicular to the axis, and 
 the segments between the 
 points ai'e bisected by it. 
 
 Corresponding points 
 are equidistant frojn the 
 axis. 
 
 Every line perpendicu- 
 lar to the axis corresponds 
 to itself, and cuts correspond- 
 ing lines in corresponding 
 points. 
 
 Points zahich join two 
 pairs of corresponding lines 
 are equidista/it from the 
 axis, and have their join 
 bisected perpendicularly by 
 the axis. 
 
 The join of two cor- 
 I'esponding points lies in 
 the centre, 
 
 for it corresponds to itself. 
 Two corresponding points 
 are equidistant from the 
 centre ; their join passes 
 through the centre ; or — 
 
 The segment between 
 corresponding points is bi- 
 sected by the centre. 
 
 The join of two cor- 
 responding lines does not 
 exist ; they are parallel, 
 for they make equal angles with 
 every line through the centre. 
 
 Corresponding lines are 
 equidistant from the centre ; 
 that is, the perpendiculars to 
 them from the centre are equal 
 in length. 
 
 Every line through the 
 centre cuts corresponding 
 lines in corresponding 
 points. 
 
 Lines which join two 
 pairs of corresponding 
 points are parallel. 
 
Examples of Symmetry. 
 
 97 
 
 To a proposition re- 
 lating to lines or angles 
 in axial symmetry we 
 have a proposition relating 
 to points or segments in 
 central symmetry. 
 
 To a proposition re- 
 lating to points or segments 
 in central symmetry we 
 have a proposition relating 
 to lines or angles in axial 
 symmetr}'. 
 
 Fig. 
 
 § 154. Examples of Symmetry. — The present and 
 the following paragraphs contain important examples 
 of symmetry in simple figures. 
 
 If a figure consists of a single 
 point or of a pencil of lines (fig. 
 68), then this point may be con- 
 sidered as a centre of symmetry, 
 and any line s through it as an 
 axis of symmetry. In the latter 
 case any two lines a, a' equally 
 inclined to the one chosen as 
 axis are corresponding lines. 
 
 If a figure consists of two points A, B (or pencils) 
 (fig. 69), then their mid point 
 M\^ 2. centre of symmetry, and 
 the two points correspond one 
 to the other. 
 
 The join s of the two points, 
 on the other hand, is an axis, 
 each of the given points now 
 corresponding to itself. The 
 line perpendicular to the join 
 and passing through the mid 
 
 point is a second axis, the two given points again 
 corresponding the one to the other. This line is 
 
 H 
 
 Fig. 6q. 
 
98 Elements of Plmie Geometry. 
 
 called the perpendicular bisector of the segment A B 
 determined by the two points. 
 
 § 155. If a figure consists oi 3iSmgle line, or 7'07u of 
 points, then this line may be considered as axis of 
 symmetry and every point in it as caitre. In the 
 latter case any two points equidistant from the one 
 chosen as centre are corresponding points. Further, 
 every perpendicular to the given line is an axis of 
 symmetry. 
 
 If a figure consists of two intersecting lines, then 
 each of the two bisectors of the angles formed by the 
 two lines is an axis of symmetry. These lines are at 
 right angles to each other (§ no). With regard to 
 each of them as axis, the two given lines correspond 
 the one to the other. 
 
 If the two given lines are at right angles, not only 
 the bisectors of their angles, but either of the given 
 lines, may be taken as axis. In this case each line 
 corresponds to itself 
 
 The join of any two intersecting lines is a centre 
 of symmetry, every line corresponding to itself 
 
 In all these cases the reader should make figures 
 for himself 
 
 § 156. If the two lines a and b are parallel they 
 have common perpendiculars. Every one of these 
 perpendiculars is an axis of 
 symmetry, for it is an axis 
 for each of the given lines. 
 If one of these perpendicu- 
 lars cuts the given lines at 
 A and B (fig. 70), then 
 the mid point O of A B is a centre of symmetry (§ 123). 
 Hence the figure lias an infinite number oi centres. 
 
 
 Fig. 70. 
 
 
 ft 
 
 A 
 
 C 
 
 
 
 
 
 - 
 
 Q. 
 
 /, 
 
 
 
 u 
 
 M 
 
 D 
 
Examples of Syinmetry. gg 
 
 If OQ be the perpendicular bisector of A B, 
 then it is parallel to the given lines a and i^, as all 
 three are perpendicular to AB. If we fold the plane 
 over along this Hne O Q, then B will fall on A and 
 BZ) on A C, siS both lines are perpendicular to OA. 
 Hence the line O Q is an axis of symmetry. As this 
 line divides the strip of the plane between the two 
 parallels a and I? into two congruent parts, it will be 
 called the bisector of the strip between the parallels. 
 Accordingly two parallel lines have the bisector of the 
 strip between them, and also every perpendicular to 
 it, an axis of symmetry, whilst every point in the 
 bisector of the strip is a ce7itre of symmetry. 
 
 § 157. A figure consisting of a line and a point 
 without it has no centre of symmetry, for a point 
 which shall be a centre of the line must lie on the 
 line, whilst the given point has only itself as a centre. 
 But the figure has an axis. Every line through the 
 point is an axis of the point, and every line perpendi- 
 cular to the line is an axis of the line ; hence the 
 perpendicular from the point to the line is an axis for 
 each, and therefore for the figure. 
 
 § 158. Two Axes of Symmetry. — In several of 
 these figures we have found two axes of symmetry at 
 right angles to one another. Thus two points or two 
 intersecting lines have two axes perpendicular to one 
 another. In each case the join of these axes is a centre. 
 In the case of two parallels we found an infinite 
 number of parallel axes, and another axis perpendicu- 
 lar to them. Here, again, every join of two of "these 
 axes is a centre. It is easily proved that this property 
 always holds, whence we have the following theorem : 
 
 H 2 
 
100 Elements of Plane Geometry. 
 
 Theorem : If a figu7'e has two axes of symmetry 
 at right angles to one another^ the?t the Join of these axes 
 is a centre of syi?imetry. 
 
 For if ^ and j^ be two axes at right angles (fig. 71), 
 then to a point A will correspond a point A' with regard 
 to X as axis. To these will correspond points A^ and 
 Ax with regard to y as axis. These points A^ and^/ 
 will correspond to each other with regard to x. To 
 see this, let us first fold over along y ; then A falls on 
 Ax and A' on A^'. If we now, without folding back, 
 fold over along x, A' and 
 with it ^1' will fall on A, 
 which coincides with A^. At 
 
 Fig 
 
 71- 
 
 
 y 
 
 A,-,^ 
 
 .,A 
 
 j ^^"v 
 
 
 .4!,'" 
 
 ^^^A' 
 
 coincide, so that the angles 
 ASx and A^ Sx' are equal, 
 where x' denotes the continua- 
 tion of X beyond S. It follows 
 that AS Ax are in a Hne, and that the segment A Ax 
 is bisected at -5* ; or 6* is a centre of symmetry for 
 A Ax', and similarly for Ai and A'. 
 
 Exercises, 
 
 In axial symmetry prove 
 that: 
 
 ( 1 ) Lines j oining any point 
 on the axis to two con-espond- 
 ing points are corresponding 
 lines. 
 
 (2) Every line perpendicu- 
 lar to the axis cuts corre- 
 sponding lines in correspond- 
 ing points. 
 
 (3) Two lines which pass 
 
 In central symmetry prove 
 that: 
 
 (i') Points in which any 
 line through the centre cuts 
 corresponding lines are corre- 
 sponding points, 
 
 (2') Parallels through cor- 
 responding points are corre- 
 sponding lines. 
 
 (4) Two lines which pass 
 
Exercises. lOi 
 
 through two corresponding through two coiTesponding 
 points A, A', and make equal points A, A', and make equal 
 but opposite angles with ^ ^', angles with A A', are corre- 
 are corresponding lines. sponding lines. 
 
 (5) In axial or central symmetry if A, A' and B^ B' are 
 pairs of corresponding points, the lines AB and A' B', as well 
 as AB' and A' B, are corresponding lines j and if a, a' and b^ b' 
 are pairs of corresponding lines, the points ab and a' b\ as well 
 as a b' and a! b, are corresponding points. 
 
 (6) Perpendiculars to the limits of an angle meeting these at 
 equal distances from the vertex intersect on the bisector of the 
 angle. 
 
 (7) If on two parallels through the ends of a segment A B two 
 points C, D be taken such that AC = DB, the line CD will 
 bisect A B. 
 
 (8) Two equal and opposite segments AB and A' B' lie on 
 parallel lines. Determine a point S such that A and A', as well 
 as B and B', are s)tti metrical with regard to S as centre. 
 
 (9) If two triangles ABC and A' B' C have the sides of 
 the one parallel respectively to those of the other, but of opposite 
 sense, if further one side in one is equal to the parallel side in 
 the other, then the three lines which join corresponding vertices, 
 viz. those which lie opposite to parallel sides, meet in a point, 
 which is the mid point of each. 
 
 (10) State and prove an analogous theorem about axial 
 symmetry. 
 
 DRA WING. 
 
 In the following constructions of symmetrical figures, it is 
 supposed that either the axis or the centre of symmetry are given. 
 
 (il) To a given point A find the corresponding point A'. 
 Solution in Axial Syttunetry. Sohition in Central Symmetry. 
 
 In the perpendicular drawTi The point A' lies on the 
 
 from A to the axis and cutting line joining A to the centre 
 the latter at M determine A' S, making A S=SA'. 
 so \h^tAM=MA'. 
 
 (i2) To any given line a find the corresponding line a'. 
 
102 Elements of Plane Geometry, 
 
 Solution in Central Symmetry. 
 Take any point A in a and 
 determine its corresponding 
 point A'. The line through 
 A' parallel to a will be the 
 required line a. (§ 153, <5.) 
 
 Solution in Axial Symmetry. 
 
 To any two points A, B in 
 a determine the corresponding 
 points A\ B'. Their join 
 gives a' . 
 
 When possible take B 
 where a cuts the axis. 
 
 If a is parallel to the axis 
 draw a! parallel to it through 
 A'. 
 
 (13) By aid of the set squares draw a pair of lines such that 
 a given line may bisect one of the angles included between 
 them (or so that the given line may be an axis of symmetry). 
 
 Solution is possible in eleven different ways. [Chap. VI. 
 Ex. (17.)] 
 
 (14) To a given point A find the corresponding point A' by 
 aid of set squares only. 
 
 Solution in Axial Symraetry. 
 
 Draw a pair of correspond- 
 ing lines /, /' [exercise (13)] of 
 which / passes through the 
 point A. Then the point A' 
 will be the point where the 
 line /' cuts the perpendicular 
 from A to the axis. 
 
 (15) Having given the axis or centre of symmetry and one pair 
 A", K' of corresponding points, coiistruct a figure symmetrical 
 to a given figure by aid of a straight-edge only, using neither a 
 pair of compasses nor a set square. 
 
 Solution in Central Symmetry. 
 
 Join A to the centre S and 
 draw a line p perpendicular to 
 AS through S. Find (as in 
 opposite column) the point A' 
 corresponding to A Avith regard 
 to p as axis. This will be 
 the required point. 
 
 Solution in Axial Symmetry. 
 
 A being any point in the 
 figure, draw AK and A IC 
 cutting the axis in Fand V. 
 The lines joining these two 
 
 Solution in Central Symvuiry. 
 
 A being any point in the 
 figure, draw through A'' a line 
 parallel to A K. This cuts the 
 line A S at A'. Or : Through 
 
Exercises. 103 
 
 points crosswise to K and K', K' draw a line parallel to 
 
 \\z. K V and IC V, meet in AK, and through K a line 
 
 the point A', corresponding parallel to AK'. They meet 
 
 to A. in A' . 
 
 To test the accuracy of your drawing see whether the 
 axis or the centre bisects the segment A A'. 
 
 (16) To a given triangle ABC construct the corresponding 
 triangle A' B' C, both in axial and central symmetry. 
 
 First by method in exercise (ii). 
 
 Second by finding A' as before and the rest by the method in 
 exercise (15). 
 
 Check each construction by the other. 
 
 (17) To a four -point find the conresponding four-point with 
 regard to a given axis or a given centre. 
 
 Determine in each figure the triangle formed by the three 
 joins of opposite sides. 
 
 Check your construction. 
 
 (18) Draw a figure s}-mmetrical to a triangle with regard to 
 one of its sides as axis. 
 
 There are three cases, according as each of the angles 
 adjacent to the side chosen as axis is acute, or one is acute and 
 the other right or obtuse. 
 
 (19) Draw the figure s>Tnmetrical to a triangle with regard to 
 the perpendicular bisector of one of its sides as axis. 
 
 {20) Draw the figure S)Tnmetrical to a triangle with regard 
 to one of its vertices, and also with regard to the mid point of 
 one of its sides, as centre. 
 
 (21) Draw the figures symmetrical to a triangle with regard 
 to the mid point of each side. 
 
 (22) Hadng given two corresponding points A and A', find 
 the axis of s}Tnmetry by aid of set squares. 
 
 Solution : Through A and A' drav/ two lines making equal 
 but opposite angles -with A A'. The perpendicular from their 
 join to A A' will be the axis required. 
 
 (23) Bisect a given segment by aid of the set squares. 
 
 (24) Find the centre of s}Tnmetry, having given two corre- 
 sponding points. 
 
104 Elements of Plane Geometry. 
 
 (25) Find an axis of symmetry so that two given lines a 
 and ci correspond to each other. (This cannot be done by aid 
 of set squares only.) 
 
 Solution : From the join V of the given lines set off equal 
 segments VA on a and VA' on a' . The perpendicular from 
 VtoAA' is an axis of symmetiy. The parallel to A A' through 
 V is another axis. 
 
 (26) Determine the bisector of any given angle, and also the 
 two bisectors of the angles made by two intersecting lines. 
 
 (27) Draw the bisector of an angle included by two lines 
 which meet off the drawing paper. 
 
 What becomes of the bisector in this constniction if the 
 lines are parallel? 
 
 (28) Find the centre of symmetry, having given two pairs of 
 corresponding (hence parallel) lines. 
 
 (29) Draw the figures mentioned in §§ 154 to 157. 
 
 CHAPTER VIII. 
 
 THE TRIANGLE. 
 
 §159. The Triangle. — A triangle has three 
 
 vertices and three sides. Each side joins two vertices 
 
 and is opposite the third, whilst each vertex joins two 
 
 sides and is ^/^j-//^? the third. Hence 
 
 Fig. 72. • -, , 
 
 every side has a vertex opposite, 
 and every vertex has a side opposite. 
 It will be convenient to denote 
 the vertices by capital letters A^ B, C, 
 and the sides by small letters a, b, c, 
 in such a manner that a vertex and 
 the opposite side are denoted by the same letter. 
 
The Triangle. 
 
 105 
 
 Fig. 
 
 73- 
 
 Fig. 74. 
 
 Hence a Is the side opposite A^ and so on, as in the 
 figure (fig. 72) ; or 
 
 BC^a, CA=b,AB^c. 
 
 Every side has a mid point. These Avill be denoted 
 by A', B\ C, so that A' is the 
 mid point of the side opposite 
 A, or of the side a (fig. 73). 
 
 A Hne which joins a vertex 
 to the mid point of the opposite 
 side is called a median li?ie. 
 Hence a triangle has three 
 median hnes, A A', BB', CO. JC 
 
 § 160. Triangle with Axis of Symmetry. — Let us 
 now see whether triangles exist which are symmetrical 
 either with regard to an axis or with regard to a centre. 
 
 If s is an axis of symmetry, 
 A any point on it, and B and 
 C any two points corresponding 
 with regard to the axis, then 
 the triangle ABC will have s 
 for an axis of symmetry. If 
 this axis cuts the side B C 2X 
 A', then A^ is the mid point of 
 B C, or A A' is a. median line. 
 
 Hence a trimigle may have a 
 median line as axis of symmetry. 
 
 This is the only case possible ; for if a triangle 
 ABC has an axis of symmetry, then to every vertex 
 must correspond a vertex. Hence if B corresponds 
 to C the third vertex must correspond to itself, or 
 it must lie on the axis. But the axis bisects the join 
 oft^vo corresponding points B and C, and is therefore 
 necessarily a median line. 
 
io6 Elements of Plane Geometry. 
 
 If we try in the same manner to construct a triangle 
 having a centre of symmetry, we do not succeed. For 
 to a side there corresponds either another side, which 
 in that case is parallel to the first (§ 153, 3), or the side 
 corresponds to itself and passes through the centre. But 
 a triangle cannot have two sides parallel, nor can it have 
 all its sides passing through the same point. Hence 
 
 A triangle cannot have a ceiitre of symmetry. 
 
 § 161. A triangle which has an axis of symmetry 
 is called a symmetrical triangle^ or an isosceles triangle. 
 We shall use only the former name. That vertex 
 through which the axis passes is called the vertex^ 
 the opposite side is called the base^ the remaining sides 
 are called simply the sides^ and the segment on the axis 
 between the vertex and the base is called the altitude^ 
 of the triangle. 
 
 The principal properties of a symmetrical triangle 
 follow immediately from the construction in § 160. 
 The one vertex A (fig. 74) lies on the axis of sym- 
 metry ; the others correspond to each other. Hence 
 the lines A B and A C are corresponding lines ; there- 
 fore they are equal. The same holds for the angles 
 at B and C, for the angles at A^ and for those at A'. 
 
 These properties may be stated thus : 
 
 Theorem : A symmetrical t7ia7igle has 
 
 I St. Two sides equal, viz. those which meet at the 
 vertex. 
 
 2nd. Two angles equal, viz. those which are opposite 
 the equal sides. 
 
 3rd. A median line bisects the angle at the vertex. 
 
 4th. A median line is perpendicular to the base. 
 
 5th. The peipendicular bisector of the base passes 
 through the vertex. 
 
The Symmetrical Triajigle. 107 
 
 § 162. Evety one of these properties conditions 
 the others, for it can in each case be proved that the 
 triangle has an axis of symmetry. The following are 
 the more important cases : — 
 
 Theorem : A triajigle is sy??tmefrical, and has 
 therefore all the properties stated in § 161 — 
 
 I St. If it has two equal sides. For the bisector 
 of the angle contained by these sides is an axis of 
 symmetry. 
 
 2nd. If it has ttuo equal angles. For the perpen- 
 dicular bisector of the side adjacent to these angles 
 is an axis of symraetrj'. 
 
 3rd. If a median line is perpendicular to the side 
 which it bisects. 
 
 4th. If the perpendicular bisector of a side passes 
 through the opposite vertex. 
 
 In the last two cases the perpendicular bisector 
 of one side passes through the opposite vertex. It is 
 therefore an axis of S}-mmetry of that vertex, and also 
 of the end points of the side which it bisects, hence 
 of the whole figure. 
 
 § 163. The Regular Triangie. — If a triangle has all 
 three sides equal, each of the three median lines is 
 an axis of symmetry, and all angles are equal. Con- 
 versely, if a triangle has all angles equal it has all its 
 sides equal. 
 
 Such a triangle is called an equilateral o^c equiangular, 
 or better a regular, triangle. 
 
 § 164. Let us now suppose a triangle ABC {i\g. 
 75) in which the bisector of the angle at A is not an 
 axis of symmetry. Then the contra-positive form of 
 the theorem of § 162 tells us that ^^ is not equal to 
 
lo8 
 
 Elements of Plane Geometry. 
 
 A C, that the angle B is not equal to the angle C, and 
 that the bisector AD oi the angle at A is not per- 
 pendicular to B C, and hence, that the two angles 
 AD B and ADC slyc unequal. Between these angles 
 there exists the relation 
 
 Z^B+l.BDA=/.C+Z.CDA, 
 
 for each sum makes with half the angle at A 
 an angle of continuation. 
 Hence it follows that, if angle 
 B is greater than angle Q 
 angle BD A is less than angle 
 CDA. 
 
 If we now fold the figure 
 along AD, then AB will fall 
 along A C ; and B will fall 
 between A and C if we sup- 
 pose that AB is the shorter of the two unequal lines 
 A B and A C. The line D B therefore takes the posi- 
 tion DB' within the angle ADC. But the angle 
 AB' D, which is equal to angle B, is exterior to the 
 triangle D CB', and therefore greater than the angle 
 at C (§ 138). 
 
 Conversely, if the angle ADB<ADC, the line 
 DB will fall within the angle ADC, and therefore B 
 will fall between A and C, that is, AB will be less 
 than A C. This always happens (see above) if 
 LB> lC, for then BDA<ADC. 
 
 Theorem: I?i every triangle the greater side is 
 opposite to the greater angle, and conversely, the greater 
 angle is opposite to the greater side. 
 
 In a right-angled triangle, for instance, the hypo- 
 thenuse is greater than either side. 
 
The Un symmetrical Triangle. 
 
 109 
 
 § 165. If we join B B' (fig. 75) we get a triangle 
 BB'C, in which lB>BB'=z lDB' B ; 
 therefore i_CBB' < CB'B ; 
 therefore ^'C<^C 
 
 But B' C=A C—A B, Hence 
 Theorem : Li every triangle the difference of two 
 sides is less than the third side. 
 
 AC-AB<BC. 
 From this it follows at once that 
 
 AB + BOAC. 
 
 Theorem : In every triangle the sum of two sides is 
 greater than the third side. 
 
 The proof given for this fundamental theorem 
 required that the two sides A B and A C should be 
 unequal. But if they are equal the theorem does 
 not require a proof, as a little consideration will 
 show. 
 
 § 166. From the last theorem we can draw an 
 important conclusion relating to inequalities in axial 
 symmetry. 
 
 Let A and A' (fig. 76) be two points correspond- 
 ing with regard to the line s as 
 axis of symmetry, and let ^ be 
 any other point on the same 
 side of the axis as A. Then 
 the hne joining B to A' will 
 meet the axis at a point C, so 
 that AC=CA'. We have 
 therefore BC-^CA' or B A' 
 = BC + CA. But BC + CA 
 >BA\ therefore BA'>BA. 
 pressed as follows : 
 
 Fig. 76. 
 
 This may be ex- 
 
no Elements of Plane Geometry. 
 
 Theorem : Of two points A A' correspo7iding with 
 regard to an axis of symi?iet7'y, that o?te is the nearer to 
 any point B which lies on the same side of the axis as B. 
 Conversely : If a poi?it B is at a shorter distance 
 fro77i A than from the point A' which corresponds to A 
 with regard to a7i axis, then A and B lie on the same 
 side of the axis. 
 
 As a corollary we get 
 
 Of two sides i7i a t7'iangle that is the g7'eater which 
 is cut by the perpe7idiciilar bisector of the third side. 
 
 § 167. Obliques. — From a points (fig. 77) without 
 a line / only one perpendicular can be drawn to tliat 
 
 line. \i AO'i^ the perpen- 
 dicular, then every other 
 line A Q through A cuts 
 the line/ under an acute 
 angle, as the triangle 
 A O Q has a right angle 
 at O. Such a Hne A Q is 
 called an oblique. 
 
 As the angle at O is greater than the angle at Q, 
 it follows that AQ>AO. Or 
 
 Theorem : Of all seg7nents draw7i to a line fro7n a 
 poi7it without it, that 07i the perpendicular is the 
 shortest. Its le7igth is called the distance of the point 
 A fro7n the line p. (In geometry by distance the 
 shortest distance is always meant.) 
 
 § 168. If from A (fig. 77) two obliques be drawn 
 to a line /, say A Q and A J^, which lie on opposite 
 sides of the perpendicular A O, and which cut off 
 equal distances, QO, OR, on / ; then A Q=AR, as 
 the triangle A QR is symmetrical with regard to A O 
 
Obliques. Ill 
 
 as axis of symmetry. It follows that from a point A 
 without a line two equal segments can be drawn to 
 the line. If, on the other hand, two obliques A Q 
 and AR are equal, then A QR is a symmetrical 
 triangle j hence the perpendicular from A to the line 
 p must bisect the side QR^ and therefore the two 
 equal obliques cut off equal segments (2(9, OR. 
 
 It follows that 
 
 Theorem : Fi'oin a poi?it A without a line two, and 
 only two, obliques which are equal to a given seg7nent 
 can be drawn to that li?te, provided that the given seg7fie?it 
 be greater tha?i the distance of the point f?vm the line. 
 
 Or, in other words 
 
 Theorem : On a line two, and only two, poi?its exist 
 which have a given dista?ice from a fixed poifit, provided 
 that the given distance is greater than the distance of the 
 point fro7n the line. These two points are symmetrical 
 with regard to the perpendicular from thepoi?it to the line. 
 
 On a line there exists one point only lohich has its 
 distance from a given point A equal to the distaitce of A 
 from the line: it is the foot of the perpendicular fvm A 
 to the li?ie. 
 
 There is no point on the liiie which has its distance 
 from a point A less than the distance of the line from 
 that poi?it. 
 
 Exercises. 
 
 (i) If a right-angled triangle is symmetrical, then the axis 
 of symmetry bisects the right angle. 
 
 (2) An angle in a triangle will be acute, right, or obtuse 
 according as the median line through the vertex of that angle is 
 greater than, equal to, or less than half the opposite side. 
 
 (3) A median line of a triangle will be greater than, equal to, 
 
112 Elements of Plane Geometry. 
 
 or less than half the side it bisects, according as the angle oppo- 
 site to that side is acute, right, or obtuse. 
 
 Show that this theorem is a logical consequence of its 
 converse, which is contained in the last exercise. 
 
 (4) On a given line where is the point whose distances from 
 two fixed points, not on the line, have the least possible sum? 
 
 Where is the point whose distances from the two fixed 
 points have the greatest possible difference? 
 
 The distances in both cases are to be taken without regard 
 to their sense. How ought the question to be worded if account 
 were taken of the sense of the distances ? 
 
 N.B. In each of the last two questions two cases at least 
 will have to be considered, inasmuch as the fixed points may be 
 on the same side or on different sides of the line. 
 
 (5) The distances of the extremities of the base of an isosceles 
 triangle from the opposite sides are equal to one another. 
 
 (6) Through the vertex A of a. triangle ABC a straight 
 line XY is drawn perpendicular to the bisector of the angle A. 
 Prove that if M is any point on X V, the perimeter of BMC is 
 greater than that of AB C. 
 
 (7) Prove that the three axes of symmetry of a regular 
 triangle meet in a point. 
 
 (8) The sum of two sides of a triangle is greater than the 
 sum of the segments joining a point within the triangle to the 
 end points of the third side. 
 
 (9) The perimeter of any convex polygon is less than that 
 of any other polygon by which it is completely surrounded. 
 
 (10) The sum of the distances of the vertices of a triangle 
 from any point within its area is less than the sum of the sides 
 {ox perimeter) of the triangle, but greater than half that sum. 
 
 (11) The sum of any two sides of a triangle is greater than 
 twice the concurrent median line. 
 
 (12) The sum of the three median lines of a triangle is less 
 than the sum of its sides, but greater than half this sum. 
 
 (13) Through a point A three half-rays a, b, and q are drawn, 
 a and b being on the same side of q, and making acute angles 
 with it. Prove that of the rays a and b that has the greater 
 
Symmetrical Quadrilaterals. 113 
 
 distance from any point B in q wliich makes the greater angle 
 with q. 
 
 State and prove also the converse theorem, and investigate 
 the case when a and b are on opposite sides of q. 
 
 (14) If two half-rays are symmetrical with regard to a line q 
 as axis, they are equidistant from any point B in q. 
 
 To this theorem write down the contra-positive, the converse 
 and its contra-positive, and prove one of the latter two. 
 
 Discuss the logical connection between this and exercise {13). 
 
 • DRA V/ING. 
 
 The following constructions should be made as far as pos- 
 sible by set squares only, without using a pair of compasses. 
 
 (15) Construct an isosceles triangle, having given 
 
 1. Half the angle at the vertex and the length of either the 
 side, or the altitude, or the base. 
 
 2. The angle at the vertex and the length either of the side, 
 or the altitude, or the base. 
 
 3. An angle at the base and the length either of the side, or 
 the base, or the altitude. 
 
 4. The length of the base and the length of either the side 
 or the altitude. 
 
 (16) Construct a regular triangle, having given the length 
 either of a side or the altitude. 
 
 (17) Draw the figures to the exercises (i) to (14). 
 
 CHAPTER IX. 
 
 SYMMETRICAL QUADRILATERALS. 
 
 § 169. By a quadrilateral will be understood at 
 ] present a polygon of four sides of which no two sides 
 ' intersect : it may be convex (fig. 78), or it may have 
 ! a re-entrant angle (fig. 79). Every quadrilateral has 
 ' I 
 
 I 
 
1 14 Elements of Pla7ie Geometry. 
 
 Fig. 78. 
 
 a vertex opposite to every vertex, an angle opposite 
 to every angle, and a side oppo- 
 site to every side. 
 
 Every side is adjaccfit to 
 two angles and to two sides. 
 / ^ Every angle is adjacent to 
 two sides and to two angles. 
 
 A quadrilateral has two 
 diagonals; they join opposite 
 vertices. A diagonal may lie 
 either wthin or without the quadrilateral. 
 
 If two opposite sides are 
 parallel, the line joining their 
 mid points is called a median 
 line. 
 
 Every quadrilateral is di- 
 vided by each diagonal into 
 two triangles. 
 
 In every quadrilateral the sum of the angles isi 
 equal to four right angles (§ 140). 
 
 § 170. A quadrilateral may have an axis or a 
 centre of symmetry. 
 
 If a quadrilateral has an axis of symmetry, then toj 
 every vertex not on the axis there corresponds a vertex! 
 not on the axis. Hence the number of vertices not 
 on the axis must be even : either two or all four verticesi 
 are off the axis. Hence also the number of vertices oni 
 the axis must be even : either two or no vertices lie on< 
 the axis. All four vertices cannot lie on the axis, for 
 four points in a line cannot form a quadrilateral. 
 
 There are therefore two cases of quadrilaterals 
 which have an axis of symmetry according as two or 
 
The Kite. 
 
 115 
 
 no vertices lie on the axis of symmetry. These 
 quadrilaterals have special names. In the first case 
 the quadrilateral is called a kite\ in the second case it 
 is a symmetrical or isosceles trapezium. These have to 
 be considered separately. 
 
 §171. The Kite. — Definition: A quadrilateral 
 which has a diagottal as axis of symmetry is called a 
 
 KITE. 
 
 Let A and B be the two vertices 
 on the axis, and CC the other two. 
 Then C and C are corresponding 
 points with regard to the axis A B ; 
 therefore the line C C is perpendicular 
 to AB and is bisected by it. This 
 proves the first part of the theorem 
 on page 116. 
 
 The diagonal point D may lie 
 within or without the kite. In the first case (fig. 80) 
 the kite is convex, and every angle is less than an 
 
 angle of continuation. 
 
 In the other case (fig. 
 
 81) the figure has a 
 
 re-entrant angle at B, 
 
 and the angle CBC 
 
 in the quadrilateral is 
 
 greater than an angle 
 
 of continuation. In- 
 termediate between these we have the case w^here D 
 coincides with B. The figure is then a symmetrical 
 triangle, which thus appears as the limiting case of a kite. 
 In the two other cases the kite is divided by the 
 transverse axis into two symmetrical triangles, which 
 
 Fig. 82. 
 
Ii6 Elements of Plane Geometry. 
 
 lie on the same or on opposite sides of the transverse 
 axis as common base. 
 
 Other properties follow from the S}Tnmetry of the 
 figure if we remember that corresponding angles and 
 segments are equal. This gives the following theorem. 
 Theorem : A kite has the follozoing pi'operties : 
 I St. One diago7ial^ the axis, is the perpendicular 
 bisector of the other, zuhich ivill be called the transverse 
 axis. 
 
 2nd. The axis bisects the angles at the vertices 
 which it joins, 
 
 3rd. The angles at the end poifits of the transverse 
 axis are equal, and equally divided by the latter, they 
 being corresponding angles. 
 
 4th. Adjacejit sides which meet on the axis are equal. 
 5th. The axis divides the kite into two triangles 
 zuhich are coftgruent, with equal sides adjacent. 
 
 6th. The transverse axis divides the kite into tzvo 
 trianghs, each of which is syfn?netrical. 
 
 7th. The median lines meet on the axis and are 
 equally inclined to it. 
 
 § 172. Conditions that a Quadrilateral may be a 
 
 Kite. — Each of the seven properties enumerated in § 171 
 
 involves the others, for a?iy quadrilateral zvhich has one 
 
 , of these properties is a kite. 
 
 j.^ I St. A quadrilateral is a kite if 
 
 one diagonal is the perpendicular 
 
 \ bisector of the other. 
 
 \ For it has that diagonal as axis 
 
 \ of synnnetry, 
 '—-/^ 2nd. A quadrilateral is a kite if ' 
 ^ one diagonal bisects the angles at the ' 
 vertices which it Joins, 
 
TJie Symmetrical Trape.':i?cm. 1 1 7 
 
 For this diagonal is an axis of symmetty, the sides 
 being in pairs equally inclined to it. 
 
 3rd. If each side is equal to one of its adjacent sides. 
 
 Proof'. If AB=AD and CB=CD (fig. 83), 
 then the two triangles ABD and CBD are sym- 
 metrical, with A C as common axis of. symmetry. 
 
 4th. Jj one diagonal divides it into two isosceles 
 triangles^ or if it is made up of tiuo isosceles triangles 
 on a CO nun on base. 
 
 For the same reason as in the 3rd case. 
 
 5th. If two adjacent sides are equal, and if the 
 a?igle cofitained by them is bisected by a diagonal. 
 
 Proof: li BAC=CAD (fig. ^t,), then ^^ and 
 AD are corresponding lines; and if, further, AB=AB, 
 then B and D are corresponding points with regard 
 to ^ C as axis of symmetry. 
 
 The other cases will be left as exercises to the 
 student. 
 
 § 173. The Symmetrical Trapezium. — If a quad- 
 rilateral has an axis of symmetr}^ which is not a 
 diagonal, then the vertices correspond in pairs — • 
 say, A to A' and B to B' A and 
 B bemg taken as pomts on 
 the same side of the axis. The 
 lines A A' and BB', as lines 
 joining corresponding points, are 
 perpendicular to the axis of 
 symmetry and are bisected by it. 
 
 The two lines A A' send BB 
 are therefore parallel. They are, 
 further, opposite sides of the quad- 
 rilateral. 
 
 Jr 
 
 \ 
 
 ■" B' 
 
 /y 
 
 ■-■A 
 
 E 
 
 -r 
 
1 1 8 Elements of Plane Geometry. 
 
 The axis of symmetry thus bisects opposite sides, 
 or it is a median Hne. 
 
 Definition : A quadrilateral which has a median 
 line as an axis of symmetry is called a symmetrical 
 
 TRAPEZIUM. 
 
 Theorem : The symmetrical trapezium has the fol- 
 lowing properties : 
 
 I St. Tivo opposite sides are parallel^ and have a 
 common perpendicular bisector. 
 
 2nd. The other two opposite sides are equals and 
 equally inclined to either of the other sides. 
 
 3rd. Each angle is equal to one, and supplementary 
 to the other., of its two adjacent a^tgles. 
 
 4th. The diagonals are equal and divide each other 
 equally. 
 
 5 th. The 07ie ??iedian line bisects the a?igle betiveen 
 the two diagonals., and likewise the angle between those 
 two sides produced, lohich it does not bisect. 
 
 6th. The other median litie bisects the two diagonals, 
 a7id is parallel to the two sides which it does not bisect. 
 
 7th. The tivo median lines a4'e each the perpen- 
 dicular bisector of the other. 
 
 § 174. Conditions that a Q,uadrilateral be a 
 Symmetrical Trapezium. — The converse of each of 
 these propositions, with the exception of the last two, 
 holds — that is, any quadrilateral which has one of the 
 first five properties is a symmetrical trapezium. 
 
 Theorem : A quadrilateral is a symmetrical trape- 
 zium : 
 
 I St. If tivo opposite sides have a common perpen- 
 dicular bisector. 
 
 For this bisector is an axis of symmetry. 
 
Syvnnetrical Trapezium. 1 19 
 
 2nd. If two sides are parallel and the afigles 
 adjacent to either of them equal. 
 
 Proof \ Through the mid point E (fig. 85) of 
 one of the parallel sides AB draw a perpendicular 
 EF to it. This \\ill be an axis of symmetry for the 
 two parallels ; also for the points 
 A and B. The lines B C and '"^ '" 
 
 AD, being equally indined to '^f^. 
 AB, will be corresponding lines. / 
 Therefore C and D are corre- j,--''' ' '''-^\ 
 
 sponding points. Hence ^i^ is ^ ^ c 
 
 axis of symmetr}' of the quadrilateral. 
 
 3rd. If two opposite sides are equal and include 
 equal angles with one of the remaining sides. 
 
 Proof'. As in the second case. 
 
 4th. If each angle is equal to one and supplementary 
 to the other of its adjacent angles. 
 
 Proof : As in the second case. 
 
 5 th. If one median line bisects an angle between the 
 two diagonals. 
 
 Proof'. Let the diagonals meet in G (fig. 85) ; 
 then GAB is a triangle in which the median line 
 GE bisects the angle at the vertex. Hence it is sym- 
 metrical. Similarly GDC'\% symmetrical, having 6^^ 
 as axis. Both triangles, therefore, have EF SiS axis. 
 
 6th. If the diagonals a?'e equal and divide each 
 other equally, so that the segments of the one are equal 
 to those of the other. 
 
 Proof iS^g. 85): \i AG=.BG, GD^GC, then 
 the bisector of the angles AGB and D G C is an 
 axis of symmetr}^ 
 
 § 175. Quadrilateral with Centre of Symmetry, 
 — If a quadrilateral has a ce?it?-e of symmetry, then to 
 
120 Elements of Plane Geometry. 
 
 each side which does not pass through the centre 
 corresponds another side. Hence the number of sides 
 which pass through the centre must be even. 
 
 We do not, however, get different figures if we 
 
 suppose that sides of the quadrilateral either do or 
 
 do not pass through the centre. For let A^ A', B, and 
 
 B' (fig. 86) be the four vertices, 
 
 j^ ' ' -Q. A corresponding to A' and 
 
 / ^\^ -^-"-""""/ B\.o B' ; then the lines A A' 
 
 j ,..--'" s~"\ / and B B' pass through the 
 
 centre and are bisected by it. 
 
 If we now join AB, BA', A'B', and B'A, we get 
 a quadrilateral with a centre of symmetry, with those 
 four hnes as sides, and with the Unes A A' and B B' 
 as diagonals. Here AB and A'B' are corresponding 
 lines ; hence they are parallel ; and so are likewise 
 ^^'and^'^. 
 
 § 176. The Parallelogram. — Definition : A quad- 
 rilateral which has a centre of symmetry is called a 
 
 PARALLELOGRAM. 
 
 Theorem: A parallelogram has the follotving pro- 
 perties : 
 
 I St . Opposite sides a re pa rallel. 
 
 2nd. Opposite sides are equal. 
 
 3rd. The diagonals bisect each other. 
 
 4th. Opposite angles are equal. 
 
 5th. The median lines pass through the centre and 
 are parallel to those sides which they do not bisect. 
 
 § 177. Condition that a Quadrilateral may be a 
 Parallelogram. — The converse to each of the first four 
 propositions is true, that is to say, every quadrilateral 
 wlu'ch has one of the properties expressed in the first 
 
The Parallelogram. I2I 
 
 four propositions is a parallelogram, and has therefore 
 the other properties. 
 
 Theorem: A quadrilateral is a parallelogram. 
 
 I St. If opposite sides are parallel. 
 
 Proof'. The mid point of any transversal is a centre 
 of symmetry wdth regard to tw-o parallel lines (§ 123) ; 
 hence the mid point of a diagonal is a centre of 
 symmetry of the quadrilateral. 
 
 2nd. If two opposite sides are parallel and equal. 
 
 Proof'. \i AB (fig. 87) is equal and parallel to 
 CD, then the mid point S oi the diagonal B D is 
 a centre of symmetr).' of the 
 parallels AB and CD, B and a '' n 
 
 D being corresponding points. / 
 
 Further, as BA = CD, A and / ,.,>s'" 
 C will be corresponding points. /^-'-''' 
 Therefore 6" is a centre of sym- 
 metry of all four vertices, and therefore of the 
 quadrilateral. 
 
 3rd. If each side is equal to its opposite side. 
 
 Proof'. l^QtABCD (fig. 88) be the quadrilateral 
 AB=.CD, BC=DA. Take the perpendicular 
 bisector SP of A C, and turn triangle A CD about 
 it. LetZ) fall on D'. Then AD'=AB, as each 
 equals DC, and CD'=CB, as each equals AD, 
 Hence A BCD' is a kite (§ 172, 3). 
 
 This shows that 
 
 Fig. 88. 
 
 LACD=/iCAB, ^[^^ 
 
 each being equal to the angle / ^ 
 CAD'. The two lines AB and .v.<5 
 CD are therefore parallel. 
 
122 EIe7nents of Plane Geometry. 
 
 The quadrilateral A B CD has consequently two 
 opposite sides, AB and CD, equal and parallel. 
 Hence it is a parallelogram according to the 2nd case. 
 
 4th. If the diagonals bisect each other. 
 
 Proof: The intersection of the diagonals is a 
 centre of symmetry for the end points of each ; 
 therefore, &c. 
 
 5 th. If the quadrilateral is convex and has opposite 
 angles equal 
 
 P^'oof : As opposite angles are equal, two adjacent 
 angles are together equal to the remaining two angles, 
 and are hence equal to an angle of continuation. 
 Consequently each side is parallel to the opposite one. 
 
 § 178. We thus see that there exist three distinct 
 symmetrical quadrilaterals, two with an axis and one 
 with a centre of symmetry. The axis is either a 
 diagonal or a median line. The three cases are — 
 
 I. Kite with a diagonal as axis. 
 
 II. Symmetrical trapeziinn with a inedian line as axis. 
 
 III. Parallelogram with ce litre. 
 
 But a quadrilateral may have both an axis and a 
 centre. 
 
 This gives two cases. 
 
 § 179. The Rhombus. — A kite may have a centre, 
 that is, it may be a parallelogram. 
 
 Definition : A kite with a centre is called a rhombus. 
 
 F,G. 89. ^ rhombus (fig. 89) has 
 
 ■g therefore all the properties of a 
 
 ^^^^T^\ kite and of a i^arallelocrram. 
 
 y^ 1? _\.c ^^^^ special properties which 
 
 ^^ j "X result are 
 
 \sj^^ I. Every diagonal is an axis 
 
 ^^ of symmetry. 
 
TJie Rectangle. 123 
 
 For the diagonals of the kite now bisect each other, 
 and hence each is the perpendicular bisector of the other. 
 
 2. All its sides are equal. 
 
 Adjacent sides are equal, as each diagonal is an 
 axis. 
 
 3. Each diagonal bisects the angles at the vertices 
 which it joins. 
 
 The converse of each of these propositions holds. 
 
 The proof is left to the reader. 
 
 § 180. The Rectangle. — A symmetrical trapezium 
 may have a centre or may be a parallelogram. 
 
 Definition : A symmetrical trapeziiun with a centre 
 is called a rectangle. 
 
 The special properties of the 
 rectangle (fig. 90) are ^ t: 
 
 I. All its angles a7'e right 
 
 a?igles. For opposite angles are g| ->i< 
 
 supplementary and equal. 
 
 2. The diagonals are equal 
 and bisect each other. 
 
 3. Each median line is an axis of sy^nmetry. 
 The converse of each of these propositions holds. 
 The proof is left to the reader. 
 
 § 181. The Square. — Lastly, a 
 quadrilateral may be both a rect- , e^'" 
 
 H 
 
 — ^^S - 
 
 angle and a rhombus. 
 
 Definition : A quadrilateral 
 which is a kite^ a sy^Jimetrical 
 trapezium., and a parallelogram is 
 called a square. 
 
 The special properties of the 
 square follow from those of the rectangle and rhombus. 
 
124 Elements of Plane Geometry. 
 
 I St. All sides are equal and all a?igles are equal 
 
 2nd. Each diagonal and each ??iedtan lifie is an 
 axis of symmetry. 
 
 3rd. The diagonals are equals a7id each is the per- 
 pendicular bisector of the other. 
 
 4th. The median liftes are equal, and each is the 
 perpendicular bisector of the other. 
 
 The converse of each of these propositions holds. 
 
 The proof is left to the reader. 
 
 Exercises, 
 
 (i) The quadrilateral formed by the bisectors of the interior 
 and exterior angles at the base of a symmetrical triangle is a 
 kite. 
 
 (2) The quadrilateral formed by the bisectors of the interior 
 angles of a symmetrical trapezium is a kite with two right 
 angles. 
 
 (3) In a symmetrical trapezium the mid point of each of the 
 parallel sides is joined to the vertices in the opposite side. 
 Prove that the quadrilateral formed by the joining lines is a kite. 
 
 (4) The points in which the bisectors of the angles at the 
 ends of the transverse axis of a kite cut the sides, or the sides 
 produced, are the vertices of a symmetrical trapezium. 
 
 (5) The four lines which connect the mid points of the con- 
 secutive sides of any quadrilateral form a parallelogram. 
 
 What will be the character of this parallelogram if the 
 original quadrilateral is any one of the six symmetrical quadri- 
 laterals treated of in Chapter IX. ? 
 
 (6) The three lines two of which connect the mid points of 
 opposite sides of a quadrilateral, and the third the mid points 
 of its diagonals, are concurrent ; and each is bisected at the 
 point of concurrence. 
 
 (7) The quadrilateral formed l)y joining the mid points of the 
 sides of an isosceles triangle to the mid point of the base is a 
 rhombus. 
 
Exercises. 125 
 
 (8) Prove that a parallelogram is 
 
 1. A rectangle if its diagonals are equal. 
 
 2. A rhombus if one diagonal is either perpendicular to the 
 other or bisects one of the angles of the parallelogram, 
 
 3. A square if its two diagonals are equal, and one is either 
 perpendicular to the other or bisects one of the angles of the 
 parallelogram. 
 
 (9) All parallelograms which have two of their sides in the 
 sides of a symmetrical triangle, and one vertex in the base of the 
 latter, have equal perimeters. 
 
 (10) The sum of the distances of any point on the base of a 
 symmetrical triangle from the sides is constant. 
 
 (11) The sum of the distances of a point within a regular 
 triangle from the three sides is constant. 
 
 (12) In what direction must a ball on a billiard table be 
 struck in order that it may return to its original position after 
 rebounding from all four sides of the table ? 
 
 (13) If a figure has two axes of symmetry at right angles, 
 then Q\Qry point is connected with three others which form a 
 rectangle, and every line is connected with three lines which 
 together form a rhombus. 
 
 (14) A regular polygon of n sides has n axes of symmetr)', 
 which all pass through a common point, called the centre of 
 the polygon, and which is equidistant from the vertices and 
 also from the sides. 
 
 If n is odd each axis passes through one vertex and through 
 the mid point of one side. Any two vertices and any two 
 sides correspond to one another with regard to one of these 
 axes. 
 
 If n is even the axes are of two kinds; one half of them pass 
 each through two vertices, the other half each through the mid 
 points of two sides. Each axis of the one kind is perpendicular 
 either to one of the same, or to one of the other kind, and the 
 centre of the polygon is a centre of s}-mmetry. 
 
 \Yith regard to each pair of rectangular axes, each vertex, 
 together with three others, forms a rectangle, and each side, 
 together with three others, forms a rhombus. 
 
1 26 Elements of Plmie Geometry. 
 
 How many such rectangles and how many such rhombi 
 exist ? 
 
 (15) The rays drawn from the centre of a regular polygon to 
 the vertices form a regular pencil of n rays, that is, a pencil in 
 which the angles between consecutive rays are all equal. 
 
 The rays drawn from the centre at right angles to the sides 
 form a second regular pencil of n rays whose rays bisect the 
 angles of the first. 
 
 DRA WING. 
 
 (16) Construct the figures mentioned below, having given 
 
 1. Of a kite (a) the transverse axis and the segments into 
 which it divides the axis ; (i3) the transverse axis and the parts 
 into which it divides one of the angles ; (7) the axis and the 
 angles which it bisects. 
 
 2. Of a symmetrical trapezium (a) the length of the parallel 
 sides and the distance between them ; ()8) the length of the pa- 
 rallel sides and the angle between the other sides produced ; 
 (7) two adjacent sides and the included angle ; (5) the angles 
 between two diagonals and the segments into which one divides 
 the other. 
 
 3. Of a parallelogram (a) two sides and an angle; {&) two 
 diagonals and the angle between them. 
 
 4. Of a rectangle (a) two sides ; (j8) a diagonal and a side. 
 
 5. Of a rhombus (a) one side and an angle; (j8) the two 
 diagonals. 
 
 6. Of a square (a) a side; (jS) a diagonal. 
 
 (17) Make the figures to the different cases of the converse 
 propositions in §§ 172, 174, 177, 179, 180, 181. 
 
127 
 
 CHAPTER X. 
 
 CONGRUENXE OF TRIANGLES. 
 
 § 182. According to § 150 any two congruent 
 figures may be placed in such a p^^ 
 
 position that they are symmetrical 
 with regard to an axis or a centre, 
 This ^^ll enable us to find the 
 conditions which are sufficient 
 to ensure the congruence of two 
 figures. Thus two triangles ABC 
 and A' B' C given in the same 
 plane may be placed in a position of symmetry if 
 they are congruent; and conversely, 
 if they can be placed in such a 
 position, then they are necessarily 
 congruent. 
 
 § 183. Let us suppose the two 
 triangles ABC a,nd A' B' C to be 
 congruent, and let us apply the one 
 to the other so that A' coincides w^th A, B' with B, 
 and C with C We may then turn the triangle 
 A' B' C over along A' B' or A\B, whereby we obtain 
 a quadrilateral A CB C, which has 
 the diagonal AB as axis of sym- 
 metry, and is therefore a kite (figs. 
 92, 93), or in special cases a 
 symmetrical triangle (fig. 94). 
 
 Or we may turn A' B' C about 
 the mid point S of AB through 
 an angle of continuation, till A' 
 
 
 Fig. 94- 
 
 ^ 
 
 C 
 
 A^ 
 
 ^ 
 
 
 B 
 
 A'\ 
 
 \ 
 
 
 B' 
 
 
 ^ 
 
 ^ 
 
 , 
 
128 Elements of Plane Geometry, 
 
 coincides with B and B' with A (fig. 95). The 
 resulting figure is a quadrilateral A CB C , with a 
 centre of symmetry at S. Hence it is a parallelogram. 
 Two congruent triangles can 
 therefore always be placed in 
 either of these two positions. 
 In the first case the two triangles 
 are of opposite sense (§ 132) ; to 
 make them coincident the one 
 has to be taken out of the com- 
 1;' mon plane. In the second case 
 they are of the same sense ; they may be made to 
 coincide by merely moving the one triangle in their 
 common plane towards the other. 
 This gives the following theorem : 
 § 184. Theorem : Two triangles which are congruent 
 may always be placed in such a position that they 
 together for7P. a kite or afi isosceles triangle. If the two 
 triangles are of opposite sense this may be done without 
 taking either triangle out of the plane; bid if the two 
 triangles have the same sense, one must be taken out of 
 the plane and turned over. 
 
 Two t?'iangles which a7'e congrtiefit may always be 
 placed in such a position that they form together a 
 parallelogram. If the two triangles are of the same 
 sense this can be done without taking either out of the 
 plane ; but if they are of opposite sense one must be 
 taken out of the plane aiid turned over. 
 
 § 185. That the converse also holds follows from 
 tlie general investigation of symmetrical figures. The 
 two triangles into which the axis of symmetry divides 
 a kite, or into which a diagonal divides a parallelogram, 
 
Congruent Triangles. 129 
 
 are equal, for they are corresponding parts in sym- 
 metrical figures. 
 
 In order, then, to prove that two triangles are 
 congruent, it is only necessary to show that they may 
 be placed so as to form together a kite, a symmetrical 
 triangle, or a parallelogram. 
 
 This requires, first of all, that one side in the one 
 should equal one side in the other. Say AB=:.A'B', 
 or, using the notation in § 159, c=^c'. If we then 
 place A on A' and B on B', but make C and C fall 
 on opposite sides of AB, we have one of the figures 
 92, 93, or 94. 
 
 § 186. The quadrilateral will be a kite if ^ C 
 =A C and B C=B C (§ 172). Hence 
 
 Theorem : T7iio triangles are eongrnent if the three 
 sides of the one are equal respectively to the three sides 
 of the other. 
 
 § 187. A quadrilateral is also a kite if two adjacent 
 sides are equal, and if the angle between them is bi- 
 sected by a diagonal (§ 172). This gives 
 
 Theorem : Two triangles are congruent if two sides 
 a?id the included angle in the one are equal respectively 
 to two sides and the included angle in the other . 
 
 § 188. A quadrilateral is, thirdly, a kite, if one 
 diagonal bisects the angles at the vertices which it 
 joins. Hence 
 
 Theorem : Tico triangles are congruent if one side 
 and th£ two adjacent angles in the one are equal re- 
 spectively to one side and the two adjacent angles in the 
 other. 
 
 § 189. These are the three principal theorems 
 about the congruence of two triangles. In each case 
 
 K 
 
130 Elements of Plane Geometry. 
 
 the equality of three elements ensures the equality of 
 the remaining ones. 
 
 The cases may be stated thus : In the first case, 
 three sides ; in the second, two sides and the included 
 angle ; in the third, one side and two angles must be 
 respectively equal. 
 
 There is one other case possible where three 
 elements are equal, viz. where two sides and an 
 angle opposite one of them, in one triangle, are equal 
 to the corresponding elements in the other. This 
 case requires a different treatment. 
 
 Let us suppose we know of two triangles ABC 
 and A'B'C (figs. 96 and 97) that 
 
 AC=A'C, 
 CB=C'B\ 
 
 angle ^= angle A'. 
 
 Let us place the two triangles in such a position 
 that A' falls on A, that A' B' falls along AB, and 
 that C and C fall on opposite sides of AB ; then the 
 line A B will bisect the angle CA C, and B' will 
 fall somewhere on the line AB, but on the same side 
 of A as B. Where it falls we 
 do not know ; but we do know 
 that AB is an axis of sym- 
 metry for the lines A C and 
 A C, that therefore C and C 
 are corresponding points, as 
 A C=A C. If, therefore, we 
 join C to B, we get a line 
 C B=-CB, since they are corresjDonding lines. 
 
 But from a point C without a line A B two lines 
 
Congruent Triangles. 131 
 
 may in general be drawn equal in length to a given 
 line CB (§ 168). Of these lines CB is one. Let 
 CB^ be the other, then B' must fall either on B or 
 on By In this case, therefore, 
 we cannot assert that the two 
 triangles ABC and A'B'C 
 are congruent. They may be 
 so, or they may not. 
 
 In the two triangles A CB 
 dindACB^ the angles ABC 
 and AB^C are supplementary, c' 
 
 as the triangle B CB^ is symmetrical. Hence 
 
 § 190. Theorem : If in two triangles two sides and 
 an angle opposite to one of them are equal to the eo?'- 
 responding elements in the other ^ then the ajigle opposite 
 the second side in the one is either equal or sicpplementary 
 to the corresponding angle in the other triangle. 
 
 In the first case the two triangles are congruent. 
 
 § 191. Corollary I. : If the atigle opposite the second 
 side is a right angle, t/ie two iria7igles are always 
 congruent. 
 
 For a right angle is equal to its supplement. 
 
 Corollary II. : If in two triangles two sides and 
 the angle opposite the greater of them are equal to the 
 corresponding ele7fients in the other, the two triangles 
 are congruent. 
 
 For the greater angle is opposite the greater side j 
 hence if a'>b', A'>B' ; hence B' must be acute, and 
 its supplement, which is obtuse, cannot be an angle in 
 a triangle satisfying the given conditions. 
 
 §192. The above are important propositions. The 
 theorem § 186, for instance, tells us that if t^vo triangles 
 
132 Elements of Plane Geometry. 
 
 have the sides of the one equal to the sides of the 
 other, then the angles of the one are equal to the 
 angles of the other ; and so on for the others. 
 
 It must always be borne in mind that those angles 
 are equal which are opposite equal sides. 
 
 To these theorems we shall add a few others, 
 which relate to triangles which are not congruent. It 
 will be seen that these follow with equal facility from 
 our investigations of symmetrical figures. 
 
 § 193. If two triangles ABC and A' B' C have 
 two sides of the one equal to two sides of the other 
 — say, a-^a'simd b-=^b' ^ then we know that, if the angles 
 included by these sides are equal, i.e. if /_ C=- L C, 
 the triangles are congruent (§ 187), hence the third 
 sides are equal, that is, 
 
 if C:=C:' then ^=r'. 
 
 We further know that if the third sides are equal then 
 the triangles are congruent (§ 186), and therefore the 
 angles included by the first sides are equal ; or 
 
 if^=^, then C=C. 
 
 It follows if C is not equal to C, 
 then c cannot be equal to (f. It 
 remains to decide which is the 
 greater. 
 
 To investigate this let us sup- 
 -^A. pose that a=.a', b=l)\ and that 
 L C> L C \ and let us place the 
 two triangles in such a position 
 that the equal sides CB' and 
 CB coincide, whilst the triangles lie on opposite sides 
 of B C (fig. 98). If we now join A A' we have a 
 
Unequal Triangles. 133 
 
 symmetrical triangle A CA\ CA being equal to 
 C A\ \\\\\\ the bisector of the angle at the vertex C 
 as axis of s}Tnmetry. But the angle CA' is the 
 sum of two unequal angles ; its bisector, therefore, lies 
 within the greater (§ 102), viz. within the angle A CB. 
 Hence the point B lies on the same side of the axis 
 of s}-mmetr}' as A\ and therefore its distance from A' 
 is less than that from A (§ 166), or A' B<AB. That 
 is to say, 
 
 Theorem : If tivo triangles have two sides of the 
 one equal to tivo sides of the other ^ but the included angles 
 unequal^ then the third sides, opposite these angles, are 
 unequal, arid that is the greater which is opposite the 
 greater angle. 
 
 § 194. The converse also holds — 
 
 Theorem : If two triangles have two sides of the 
 one equal to two sides of the other, and the third sides 
 unequal, then the angles opposite these third sides are 
 unequal, and that is the greater which is opposite the 
 greater side. 
 
 The proof is the same as in the former theorem. 
 But it may also be considered as a logical conse- 
 quence of the last together with the theorem in § 186; 
 for the angles in question cannot be equal, as this 
 involves the congruence of the triangles. Neither can 
 the greater angle be opposite the smaller side. This 
 would contradict the theorem in § 193. 
 
 Exercises. 
 (i) Two triangles are congruent (a) if two sides and a median 
 line in the one are equal respectively to two sides and a corre- 
 sponding median line in the other (two cases), {&) if the three 
 median lines in the one equal those in the other. 
 
134 Elements of Plane Geometry. 
 
 (2) In each of the following cases prove that the two figures 
 named are congnient, if they have the elements mentioned re- 
 spectively equal: 
 
 1. Two kites: (a) one side and the two adjacent angles ; 
 (j8) two adjacent unequal sides and the included angle. 
 
 2. Two symmetrical trapeziums : (a) the four sides ; ()8) two 
 adjacent sides and the included angle ; (7) two adjacent sides 
 and a diagonal ; (5) the two parallel sides and their distance. 
 
 3. Two parallelograms : (a) two adjacent sides and a diago- 
 nal ; ()8) two diagonals and the included angle ; (7) two sides and 
 the included angle. 
 
 4. Two rhombi : (a) one side and one angle ; (/3) two 
 diagonals ; (7) one side and one diagonal. 
 
 5. Two rectangles : (a) two adjacent sides ; {&) a side and a 
 diagonal; (7) two diagonals and the included angle. 
 
 6. Two squares : (o) one side ; ()8) one diagonal. 
 
 (3) Two quadrilaterals A BCD and A'B'CD' are con- 
 gruent \i AB = A'B\ BC=B'C, CD= CU, DA ^ DA', 
 and if the angle at A equals the angle at A', provided that botli 
 quadrilaterals are convex, that both have a re-entrant angle, or 
 that in each one side meets another. 
 
 (4) Two regular polygons of the same number of sides are 
 congruent if a side in the one equals a side in the other. 
 
 (5) If through a row of equidistant points in a line parallel 
 lines be drawn, then every line will be cut by these parallels in 
 a row of equidistant points, and the parallels will be themselves 
 equidistant. 
 
 DI^A WING. 
 
 (6) Divide a given segment into ;/ equal parts. [See Ex. 
 
 (5).] 
 
 (7) Construct a right-angled triangle, having given (o) a side 
 and the opposite angle, ()8) a side and the segment adjacent to 
 it on the hyjiothenuse terminated by the perpendicular from the 
 vertex of the right angle, (7) the hypothenuse and the sum of 
 the two sides. 
 
 (8) Construct a regular triangle having a perimeter of given 
 length. 
 
Conditions fixing a Point. 1 3 5 
 
 (9) Construct, by aid of set squares and of a pair of com- 
 passes, regular convex polygons of three, four, six, eight, twelve 
 sides, having a side of given length. 
 
 (10) Construct out of the regular convex polygons all possi- 
 ble regular star polygons of six, eight, or twelve sides (o) by 
 diagonals, joining each vertex to the next but one, or to the next 
 but two ; ()8) by the sides, producing each side till it meets the 
 next but one or the next but two. 
 
 CHAPTER XI. 
 
 LOCI OF POINTS AND SETS OF LINES. 
 
 § 195. There are in a plane an infinite number of 
 points, and also an infinite number of lines. If it is 
 only required of a point that it shall lie in a plane, we 
 have the choice between all points in the plane. But 
 if it is required of the point that it shall lie in a line, 
 then our choice, though still lying between an infinite 
 number of points, is restricted to a much narrower 
 field. It has been reduced from a spread of points 
 of two dimensions to one of one dimension only. If 
 we likewise know that the point lies on another given 
 line, it is perfectly determined, for the two lines have 
 but one point in common. 
 
 Reciprocally, a line in a plane leaves us a choice 
 between the elements of a two-way spread. But if 
 the line in a plane is required to pass through a given 
 point A in the plane, it must be a line in the pencil 
 which has A as centre. Our choice is therefore 
 limited to a one-way spread of lines. 
 
 If we further know that the line passes through 
 
136 Elements of Plane Geometry. 
 
 another point B^ it must lie in a second pencil with B 
 as centre, and is now perfectly determined, for two 
 points have but one line in common. 
 
 Hence the condition that a point shall lie on a 
 line, and, in the reciprocal figure, the condition that 
 a line shall pass through a point, is of such a character 
 that tivo conditions of this kind fix the position of the 
 point or of the line in a plane. 
 
 § 196. Instead of saying that a point shall lie in a 
 line, we may say that the point shall be one of the 
 points in a row, and instead of saying that the lines 
 shall pass through a point, we may say that it is a ray 
 in a pencil. Two rows always have one element in 
 common unless their bases are parallel. In the latter 
 case they have in general no point in common ; but 
 they have all in common if they have one in common, 
 for then the bases coincide. 
 
 Two pencils always have one ray in common if 
 both pencils are central pencils, or if the one is a 
 central and the other a pencil of parallels. If both 
 are pencils of parallels, they have in general no 
 common ray ; but they have all rays common if they 
 have one in common, for the two pencils will in this 
 case have the same direction, and will therefore 
 coincide. 
 
 § 197. Similarly if it is recjuired that a point shall 
 lie in a given line and have a given distance from a 
 given point, we know (§ 168) there are two points 
 which satisfy both conditions. Here again two con- 
 ditions fix the position of a point. 
 
 As we proceed we shall see that there are ahvays 
 two conditions required to determine the position of 
 
Conditions fixing a Point or a Line. 137 
 
 a point or of a line, or at least to leave us only the 
 choice between a finite number of points or lines. 
 
 § 198. To resume our investigation, if one con- 
 dition onl}'- is given, there will be an infinite number 
 of points or lines which satisfy this condition. 
 
 All these points or lines will, as a rule, follow con- 
 tinuously, and will constitute a one-way spread of 
 points or lines. 
 
 This spread of points or lines forms a figure 
 which has the property that every point or line belong- 
 ing to it satisfies a given condition, and conversely 
 that all points or lines satisfying the given condition 
 belong to it. 
 
 In other words, it contains all the points, or lines, 
 which satisfy a given condition, and only such points 
 or lines. 
 
 Instead of a condition we have a figure which 
 replaces it ; and this figure is called in the one case a 
 locus of points., in the other a set of lines, satisfying the 
 given condition. 
 
 § 199. The investigations of loci of points and of 
 sets of lines are reciprocal, and may therefore be 
 conveniently treated together. This will be done, as 
 in the case of axial and central symmetry, by printing 
 corresponding propositions in columns opposite to 
 each other on the page. Thus we get 
 
 Definition : The aggre- Definition : The aggre- 
 
 gate of all points in a gate of all lines in a plane 
 plane satisfying a given satisfying a given condition 
 condition is called the lo- is called the set of lines 
 cus of points satisfying satisfying that condition, 
 that condition. 
 
138 
 
 Elements of Plane Geometry. 
 
 § 200. Example of Loci and Sets 
 will illustrate this. 
 
 Problem : Let it be 
 
 required to find a point P 
 which shall have the pro- 
 perty that the line joining 
 it to a fixed point A shall 
 pass through another fixed 
 point B^ then the point P 
 must lie in the line joining 
 AB\ and as every point 
 on this line satisfies the 
 condition, it follows that 
 the required locus is the 
 line joining A to B. 
 
 -An example 
 
 Problem : Let it be 
 required to find a line / 
 which shall have the pro- 
 perty that the point 
 joining it to a fixed line a 
 shall lie in another fixed 
 line ^, then the line / 
 must lie in the join ab\ 
 and as every line in this 
 point satisfies the con- 
 dition, it follows that the 
 required set is the pencil, 
 with centre on the join of 
 a and b. 
 
 If the two given lines 
 a and b are two distinct 
 lines which are not paral- 
 lel, this set always exists. 
 § 201. If, in the second problem, two lines a and b 
 are parallel, then the set consists of a pencil of 
 parallels — viz. of lines parallel to a and b. For the 
 line / is required to cut a where b cuts a. But b does 
 not cut a if they are parallel ; therefore / must not 
 cut a. 
 
 We see thus that the pencil of parallels appears 
 here as a special case of the pencil of lines passing 
 through a point, and that therefore the name pencil of 
 parallels is justified. 
 
 § 202. Problem a : To Problem b : To find 
 
 find the locus of points the set if lines which are 
 
 If the given points 
 A and B are two distinct 
 points, this locus always 
 exists. 
 
J? 
 
 Loci and Sets of Lines. 139 
 
 which are equidistant from equally inclined to two given 
 tivo given points. lines. 
 
 To solve the first problem let A and B be the 
 given points (fig. 99), and let P be any point satisfying 
 the condition. Then FA=FB. It 
 
 r IG. 99. 
 
 follows at once that APB is a sym- 
 metrical triangle, which has AB a.s 
 base, and B as vertex, and further that 
 the vertices of all such triangles satisfy 
 the condition. 
 
 But we know (§ 161) that the ^ 
 perpendicular from the vertex of a 
 symmetrical triangle to the base is the perpendicular 
 bisector of the base ; and as there is only one such 
 bisector, it follows that all points satisfying the given 
 condition He on the perpendicular bisector of the 
 base. We know also that every point on the per- 
 pendicular bisector of the line AB is equidistant 
 from the two points A and B, for A and B are 
 symmetrical with regard to the perpendicular bisector 
 of the segment A B. Hence 
 
 Theorem : The perpendicular bisector of the seg- 
 ment AB is the locus of poiiits equidistant fro77i the 
 two points A and B. For this line contains all points, 
 and o?ily points, which satisfy the given condition. 
 
 It is therefore the same thing whether we say that 
 a point F shall be equidistant from two given points 
 A and B, or that the point F shall lie in the perpen- 
 dicular bisector of A B. Thus we have replaced the 
 given condition by a line. 
 
 § 203. To solve the second problem, let a and b 
 (fig. 100) be the given lines, which we will suppose 
 
I40 
 
 Elements of PlaJie Geometry. 
 
 Fig. ioo. 
 
 not to be parallel, and let / be any line satisfying 
 the condition. Then l.pa=/_pb. It follows at 
 
 once, since apb has two 
 equal angles, that it is 
 a symmetrical triangle, 
 having ad as vertex and 
 J> as base, and that all 
 bases of such triangles 
 satisfy the condition. If 
 we draw through the 
 vertex of this triangle a 
 line parallel to the base 
 J>, this will bisect the angle ba, because this line also 
 will be equally inclined to a and b. It follows that 
 every line which satisfies our condition is parallel to a 
 bisector of an angle between a and b. 
 
 But also, conversely, every line which is parallel 
 to such a bisector makes equal angles with the two 
 lines, and therefore belongs to our set. Now, there 
 are two such bisectors, and these are at right angles 
 the one to the other ; hence 
 
 Theorem : T/ie set of lines equally inclined to two 
 give?i lines which are not pai-allel consists of two pencils 
 of parallels. Each pencil is parallel to one of the bisec- 
 tors of the angles between the two lines. The directions 
 of the two pencils are therefore perpendicular to one 
 another. 
 
 It is therefore the same thing whether we say that 
 a line/ shall.be equally inclined to two given lines a 
 and b^ or that the line / shall belong to a pencil of 
 parallels, parallel to one of the bisectors between the 
 angles. 
 
Loci and Sets of Lines, 141 
 
 § 204. Problem a : To Problem b : Tofi7idthe 
 
 find the locus of points set of lines which have 
 which have equal distances equal distances from two 
 from two given lines. given points. 
 
 To solve the first of these problems, the given 
 figure consists of two lines, a and ^, which in general 
 will intersect at a point — say, 
 
 ^ ■' ' Fig. ioi. 
 
 O. \i P (fig. ioi) be a point .^ ^' 
 
 satisfying the condition, then ^^^x^ 1 ^<^ 
 
 the perpendiculars PQ and I ^^^\\^y^ \ 
 PR from P to the two fines "'—c—-~fp^^:^~-—.-<^ 
 are equal. But then the Jy^ \ "^C 
 right-angled triangles OPQ^ ^ ' ^ 
 
 and O PR are congruent (§ 191), because they have 
 the hypothenuse in common, and the two sides PQ^ 
 and PR equal. Hence the angle ROP will be equal 
 to PO Q ', so the fine 6>P bisects the angle at O^ and 
 is therefore known. This bisector is an axis of sym- 
 metry, and therefore every point in it is equidistant 
 from a and b. Hence every point satisfying the given 
 condition lies on a bisector of an angle between a 
 and b, and every point in such a bisector satisfies the 
 condition. 
 
 Now, there are two lines, and only two fines, which 
 bisect the angles between a and b. It follows that — 
 
 Theorem : The locus of points equidistant fvm two 
 given lines consists of the two lines which bisect the 
 angles between the given lines. 
 
 These two lines are perpendicular to one another 
 (§ no). The locus consists here of two lines, which 
 together constitute it. Two lines which go together 
 in such a manner are generally called a line-pair. 
 
142 Elements of Plane Geometry. 
 
 If the two given lines are parallel the locus will 
 consist of one line only^ which lies half-way between 
 the giveii lines ^ and is parallel to them. It is, again, an 
 axis of symmetry for the given lines. The proof of 
 this assertion is left to the student. 
 
 It is of interest to ask, What has become of the 
 second line in this case ? The student will easily see, 
 if he considers the line m fixed, and supposes a and b 
 to turn about two points A and B symmetrical with 
 regard to m, so that the point O moves to a greater 
 and greater distance, how the line m' also moves to a 
 greater and greater distance, till at last, if a and b 
 become parallel, the point O and the line ;//' disappear 
 at an indefinite distance out of our reach. 
 
 § 205. To solve the reciprocal problem, let us sup- 
 pose that p (figs. 102 and 103) is a line satisfying the 
 condition; then the perpendiculars A A' and B B' ., 
 drawn from the given points to /, will be equal. But 
 they are also parallel, as both are perpendicular to /. 
 Hence the quadrilateral formed by the four points 
 A^ B, A', and B' is a parallelogram (§ 177), as the op- 
 posite sides A A' and B B' are 
 
 r IG. 102. ^ 
 
 ^ , equal and parallel. Of this paral- 
 
 ^ lelogram the line A B will either 
 be a side or a diagonal. 
 
 To distinguish these cases we 
 need only observe that the two 
 points must either lie on the same side of / or on 
 opposite sides. In the former case (fig. 102) AB 
 is one side and p is opposite to it, hence parallel 
 to^^. 
 
 In the other case (fig. 103) AB will be one 
 
 lA 
 
Lines Equidistant from tzuo Points. 143 
 
 diagonal and / the other ; hence p will pass 
 
 through the mid j^oint M of ^ 
 
 AB. "'■ ., 
 
 It follows that any line ^^r 
 
 satisfying the given condition ^^.-^w^ \ 
 
 is either parallel to the line \^^:^._..^/. -^3 
 
 AB or passes through the mid \ / ^-^""^ 
 point of the segment AB. X^"""^ 
 
 We conclude that ail lines ^ ^ 
 of our set are included in a pencil of lines parallel to 
 A B and in a pencil of lines passing through M, and 
 it only remains to show that ei'ejy line in these pencils 
 satisfies the condition, or that these pencils contain 
 only lines of our set. But this is easily seen. 
 
 First, if/ be any line parallel to A B, then the per- 
 pendiculars from A and B lop are parallel,and constitute, 
 together with A B and /, a parallelogram in which the 
 perpendiculars are opposite sides, and therefore equal. 
 
 Secondly, if/ be any line through M (fig. 103), 
 then the perpendiculars A A' and B B' to it form 
 with AB' and A'B 3. quadrilateral, of which M is 
 a centre of symmetr}^, because A A' and B B' are 
 parallel. Hence it is a parallelogram, and has there- 
 fore opposite sides equal, so thaX A A' == B B' . The 
 line / therefore satisfies the condition. 
 
 This shows that our set consists of two pencils, the 
 one central, with its centre at the mid point of AB^ 
 the other a pencil of parallels to the line A B. 
 
 As the solutions to our problems (§ 204) we 
 may, then, state the follo\nng two theorems : 
 
 § 206. Theorem a: T/ie Theorem b : The set of 
 
 locus of points equidista?it lines which have equal 
 
144 Elements of Plane Geometry. 
 
 from two given lines which distances from two given 
 a?'e not parallel consists of points A and B consists of 
 the two rows lahose bases a pencil of li?tes having the 
 bisect the angles between mid point of A B as centre, 
 the given lines. together with a peficil of 
 
 If the tiuo lines are lines parallel to A B . 
 parallel, the locus consists 
 of one row whose base 
 bisects the strip between the 
 two parallels. 
 
 § 207. Method of Intersection of Loci. — The in- 
 vestigation of loci and sets is of great use in all 
 problems where it is required to find points or lines 
 which satisfy two conditions. For if we leave out one 
 condition, we may find a locus of points, or set of 
 lines, satisfying the other condition. 
 
 Thus each condition may be replaced by the 
 corresponding locus or set. If these two loci, or sets, 
 have an element in common, we get points or lines 
 belonging to both loci or sets, and therefore satisfying 
 both conditions. /\nd these will be all the points or 
 lines satisfying both ; for if a point does not lie on 
 both loci, or if a line does not belong to both sets, it 
 will not satisfy both conditions. 
 
 § 208. Problem a : Problem b : Let it be 
 
 Let it be required to find required to find lines 
 points 7uhich shall have ivhich shall be equally in- 
 equal distances from two dined to tioo gii'en lines 
 given points A and B, a/ul a and b, and also be at 
 also be at equal distances equal distances from two 
 from two given lines a, b. given points A and B. 
 
 The first condition The first condition 
 
Intersection of Loci. 
 
 145 
 
 demands that the points 
 required lie on the per- 
 pendicular bisector oiAB. 
 This line we may call c 
 (fig. 104). The second 
 
 demands that the lines re- 
 quired be parallel to one 
 of the bisectors of the 
 angles between the lines 
 a and b. The second 
 
 condition gives as locus 
 the line-pair bisecting the 
 angles between the given 
 lines a, b. These two 
 lines may be called m and 
 vi'. The line c in general 
 cuts each of the lines m 
 and m' in one point, and 
 in one point only. \{ P 
 and P' are these points, 
 then they will satisfy both 
 the given conditions. Our 
 problem, therefore, has 
 two solutions ; or 
 
 There are in general 
 two, and only two^ points 
 which are equidistant from 
 two given points and also 
 
 condition requires that it 
 be either parallel to the 
 line AB ox that it pass 
 through the mid point M 
 of the segment A B. The 
 pencil M has one ray 
 in common with each of 
 the two pencils of paral- 
 lels, given by the first 
 condition. The pencil 
 of parallels to AB has 
 in general no line in 
 common with the above 
 pencils of parallels. 
 
 Hence 
 
 There are in general 
 two, and only two, lines 
 which are eqtmlly ificlined 
 
146 Elements of Plane Geometry. 
 
 equidistant from two given to two given lines, and are 
 lines. equidistant from two given 
 
 points. 
 The exceptions which are possible for special 
 positions of the points and lines are left to the 
 student to investigate. 
 
 § 209. More important results are obtained if 
 we ask 
 
 Problem a: To find Problem b: Tofindli7ies 
 points equidistant from equally inclined to three 
 tlwee given points A^B,a?id given lines a, b, and c, 
 C, which are not in a line, which do not pass through a 
 and which therefore form poi?it, and of which no two 
 a triangle. are paf-allel, and which 
 
 therefore fo7'm a triangle. 
 To answer the first question we take first the locus 
 of points equidistant from ^ and ^ (fig. 106), that is 
 the perpendicular bisector of A B, and next the locus 
 of points equidistant from A and C, viz. the perpen- 
 dicular bisector of A C. These two lines, provided 
 they are not parallel, meet in one point (9, and in 
 one point only. They are 
 parallel only if the three 
 points A, B, C lie in a line. 
 Hence 
 
 §210. Theorem : There is 
 one, and only one, point equi- 
 ' '' distant from three given points 
 
 which do not lie in a line. 
 
 This point O is equidistant from A and B, and 
 also from A and C, therefore from B and C. But all 
 points equidistant from B and C lie on the perpendi- 
 cular bisector of B C. This gives the following 
 
Properties of Triangles. 
 
 H7 
 
 Theorem : T/ie t/iree perpendicular bisectors of the 
 sides of any triangle meet in a point. 
 
 § 211. The reciprocal problem stated above has 
 no solution, and gives a new example that the 
 principle of duality has exceptions. In the higher 
 parts of geometry these exceptions are gradually 
 removed. 
 
 § 212. Problem a : To 
 find poijits equidistant front 
 
 Problem b : To find 
 lines equidistant from three 
 
 three given lines which given points which form a 
 
 form a triangle 
 
 triangle. 
 
 llG. I«7. 
 
 To solve the first problem, let a, b, c (fig. 107) 
 be the given lines intersecting in the three distinct 
 points A, B, C. The locus of points equidistant 
 from the two lines b and c consists of the two 
 bisectors of the angles between the lines. The 
 locus of points equidistant from the two lines c 
 and a similarly consists of two lines, bisecting the 
 angles between c and a. 
 Our two loci consist thus 
 of two pairs of lines. 
 Each line of the one pair 
 cuts each line of the se- 
 cond pair in one point, 
 so that we get four points \ i / 
 
 O, (9i, 0-2, 6>3 com- \ 1 / 
 
 mon to the two loci. \ ■ / 
 
 Hence XJ/ 
 
 Theorem : There are A?* 
 
 /// general four points 
 which are equidistant from the three sides of a triangle. 
 
 § 213. Every one of these four points is equi 
 
 ^A 
 
 ^p^. 
 
148 Elements of Plane Geometry. 
 
 distant from b and c^ and also from c and «, therefore 
 also from a and b. But all points which are equi- 
 distant from the lines a and b lie on the bisectors 
 of the angles between these lines. It follows that 
 the four points <?, 6>i, 6>2, 6>3 He on the bisectors of 
 the angle between a and b, and it is easily seen that 
 each of these bisectors contains two of them. Hence 
 Theorem : The bisectors of the interior and exterior 
 angles of a triangle meet four times by threes in a 
 point. 
 
 § 214. To solve the reciprocal theorem let A, B, 
 C be the three points forming a triangle (fig. 108), 
 and let A', B', C be the mid points of the sides, viz. 
 A' oi BC,B' of CA, and C of AB. Then the set 
 of lines equidistant from B and C consists of two 
 pencils of lines, the one parallel to B C, the other 
 passing through the mid point A' oi B C. Similarly 
 the set of lines equidistant 
 from C and A consists of two 
 pencils, the one parallel to 
 CA, the other passing through 
 the mid point B' of CA. A 
 _ line belonging to both sets 
 satisfies all conditions. Of 
 these lines there are three. First, the line A'B' be- 
 longs to the two pencils A' and B' ; hence it gives a 
 solution. The pencil of parallels to BC contains 
 one line through B' and gives a solution. Lastly, 
 the parallels to CA contain one line passing through 
 A'. The two pencils of parallels contain no common 
 ray. Hence 
 
 Theorem : There are three, a /id only three, lines 
 
Properties of Triangles, 149 
 
 which arc equidistant from three given points not lying 
 in a line. 
 
 § 215. Each of these three Hnes is equidistant 
 from A and B, and is therefore either parallel to AB 
 or it passes through the mid point C of A B. 
 
 Through each of the mid points A' and B' of the 
 other sides pass two lines, whilst one line is parallel 
 to each of these sides. Hence we conclude that tvvo 
 of our three lines pass through C, whilst one is pa- 
 rallel to AB. The lines parallel to the other sides 
 cannot be parallel to AB. Hence the line passing 
 through the mid points A' and B' of two sides is 
 parallel to the third ; or 
 
 Theorem : In every triangle the litie joining the mid 
 points of two sides is parallel to the thii'd side. 
 
 The three lines equidistant from three points 
 A, B, C are therefore the three sides of a triangle 
 whose vertices are the mid points of the sides of the 
 given triangle. 
 
 § 216. We shall finish this chapter with a deter- 
 mination of the number of lines which pass through 
 a given point F and which are equidistant from 
 another given point S. 
 
 If through the point F a line a be drawn, and 
 through the point S a perpendicular to this line, 
 meeting it at Q, then the points Q, S, and F will 
 form a triangle, provided that Q does not coincide 
 with either F or S. 
 
 If Q coincides with S, the line a through F must 
 pass through S. In this case the perpendicular from 
 6" to the line has no length. If Q coincides with F, 
 the line a must be perj^endicular to SF. In this case 
 
150 Elements of Plane Geometry. 
 
 the distance of S from a is equal to SF. In every 
 other case we get a triangle SQF with a right angle 
 at Q, so that the side SQ \s less than SF, the 
 hypothenuse. The distance of the line a from the 
 point 6* is therefore less than S F. 
 
 In this case the line a', symmetrical to a with 
 regard to the line SF as axis, has the same distance 
 from S as a. And this is the only line which has this 
 property ; for if we suppose that a" is a line having 
 the same distance from S as a, then S must lie on 
 the bisector of an angle between the lines a and a" 
 (§ 206), that is, the hne SF bisects the angle at^P, 
 and is therefore an axis of symmetry for a and a". 
 But there is only one line symmetrical to a with regard 
 to SFj so that a" must coincide with a'. From all 
 this it follows that 
 
 Theorem : Through a given point F one line can 
 be drawn which has no distance fivm another giveit 
 point S ; two lines whose distances from S are both 
 equal to a segment which is less than the distance SF 
 between the given points ; one line which has its dis- ' 
 tance from S equal to SF; and no line which has its 
 distance greater than SF. 
 
 Exercises. 
 
 (i) Find the locus of the mid points of segments drawn 
 from a given point A \.o Vi given line which does not pass 
 through A. 
 
 (2) Given the sum (or the difference) of the distances of a 
 point from two intersecting lines. Find its locus. 
 
 (3) Segments are drawn between the sides and parallel to 
 the base of a symmetrical triangle. Find the locus of their mid 
 ooints. 
 
 (4) The points where any two lines parallel to the base of a 
 
The Circle. 151 
 
 symmetrical triangle cut the sides are joined crosswise. Find 
 the locus of the intersection of these joins. 
 
 (5) Find the locus of points at a given distance from a given 
 line, the distance having a definite sense. 
 
 (6) Two congruent pencils have such a position that the two 
 coincident rays, one belonging to each pencil and passing through 
 the centre of the other, correspond to each other. Find the locus 
 of joins of corresponding rays. 
 
 (7) Two congruent rows are placed in such a position that 
 (o) a point in the one coincides with its corresponding point in 
 the other, or (^3) that their bases are parallel. Determine the 
 set of lines joining corresponding points. Distinguish two cases 
 (j8) according as the rows have the same or opposite sense. 
 
 (8) In a given line find a point (a) equidistant from two 
 given points, (j8) equidistant from two given lines, (7) so that 
 the lines joining it to two given points are equally inclined to it. 
 How many such points exist in each case? 
 
 (9) In a given point find a line which is (a) equally inclined 
 to two given lines, or ()3) equidistant from two given points. 
 
 (10) Find a line which shall have equal distances from two 
 given points A^ and A^, and also equal distances from two 
 other points B^ and B^, How many such lines are possible 
 (a) when the lines ^1^2 and Bi^B^. are not parallel, or (^) 
 when they are parallel ? 
 
 (11) State and solve the problem reciprocal to the last. 
 
 DRA WING. 
 
 (12) Draw the figure to each of the above exercises. 
 
 CHAPTER XII. 
 
 THE CIRCLE AS LOCUS. 
 
 § 217. Definition of the Circle. — If a segment of 
 a line turns about one of its end points, the other 
 
152 Elements of Plane Geometry. 
 
 e7id point describes a curve which is called a circle. 
 
 The fixed end point is called the centre of the circle^ 
 
 and the moving segmoit in any 
 
 Fig. 109. position is called a radius of the 
 
 circle. 
 
 § 218. As the moving line, 
 and with it the moving end point, 
 after making an entire revolution, 
 returns to its original position, it 
 follows that the circle is a closed 
 curve. It divides the plane into two parts. The one 
 is finite, and is swept over by the moving segment 
 whose end point describes the circle. Any point 
 in this part is said to lie within the circle. Any point 
 lying in the remaining part of the unlimited plane is 
 said to lie without the circle. 
 
 § 219. Any line drawn from the centre to a point 
 on the circle is a radius, for it is one of the positions 
 of the describing segment. As this segment is of 
 invariable length, we see that all radii are equal. Hence 
 all points o?t the circle have the same distance froin the 
 cetitre. This distance is equal to the length of the 
 radius, and is called the radius dista?tce. 
 
 All points at the radius distance from the centre 
 are on the circle ; for the line joining such a point 
 to the centre is a radius of the circle, and the 
 moving segment in describing the circle coincides 
 with it once. 
 
 77ie circle is therefore the locus of points equidistant 
 from a fixed pointy the centre. 
 
 Any point R within the circle lies on some radms 
 SP, and therefore SR<SP. 
 
Fundamental Properties of the Circle. 153 
 
 Similarly if (2 is a point without the circle, then 
 SQ contains a radius SP, so that SQ>SP; or, 
 
 Theorem : .A point lies within, on, or without a 
 circle, according as its distance from the centre is less 
 tha?i^ equal to, or greater than the radius distance of 
 the circle. 
 
 § 220. Fundamental Properties of the Circle — 
 Two circles with equal radii are equal, and can be 
 made to coincide if the centre of the one he placed o?i 
 the centre of the other. They are called equal circles. 
 For every point on the second circle is in this position 
 at the radius distance from the centre of the first, and 
 therefore on the first. 
 
 The second circle may, in this position, be turned 
 about its centre, and still it will coincide unth the fist. 
 Hence also a circle ca?i be made to slide alo7ig itself by 
 being turned about its ceiitre. For of the above coin- 
 cident circles we may consider the one simply as a 
 trace of the other. This property of the circle, that 
 it is a cur\-e which can slide along itself, is its fun- 
 damental property. It allows us to turn any figure, 
 connected with the circle, about the centre without 
 changing its relation to the circle. We shall often 
 make use of it. 
 
 § 221. Theorem : A line cannot have more than 
 two points in common with a circle. 
 
 For there are (§ 168) never ^^^- "o- 
 
 more than two points in a line 
 which have their distances from 
 any fixed point, here the centre, 
 equal to a given length, here 
 the radius distance. 
 
154 Elements of Plane Geometry. 
 
 § 2 22. Definitions: A line a which cuts a circle at 
 two points A^ B (fig. no) is called a secant of the 
 circle. The segment AB on this line, which is 
 bounded by the circle, is called a chord of the circle. 
 
 A chord through the centre is called a diafnetcr 
 (CZ>infig. no). 
 
 § 223. Properties of Diameters. — Every diaineter 
 is bisected by the centre of the circle. Every dia?netcr is 
 eqnal to two radii. 
 
 All diameters are eqnal. 
 A circle caiinot have more than one centre. 
 For if it had two, the line joining them Avould be 
 a diameter having two mid points. 
 
 § 224. Centre of Symmetry. — A centi-eofthe circle 
 is a cent7'e of synwietry, the end poi7its of any diameter 
 being corresponding points. 
 
 This follows at once from the fact that the circle 
 slides along itself when turned about its centre. 
 
 § 225. Axes of Symmetry. — Every diatjiettr is an 
 axis of symmetry. For if we fold over along a diameter 
 d^ every point on the part of the circle turned over 
 must fall on some point on the 
 other, as it is at the radius dis- 
 ance from the centre which 
 remains fixed. 
 
 Conversely, Every line which 
 
 is an axis of symmetry of a 
 
 circle is a diatneter of the circle. 
 
 For, if not, there would be a point symmetrical to the 
 
 centre, and this too would again be a centre. Hie circle 
 
 would thus have two centres, which is contrary to § 22^^. 
 
 The circle has therefore an infinite number of axes 
 
Arcs of a Circle. 155 
 
 of symmetry, and has besides a centre of symmetry. 
 This, together with the property stated in § 220, that 
 the circle coincides with itself after turning about the 
 centre through any a?igic, allows us to state at once a 
 great number of its properties. 
 
 § 226. Definitions: Any two points ^,^ (fig. 112) 
 on the circle divide it into two parts, 
 which are called a7rs. If a given point, 
 in describing a circle, moves from one 
 position A to another B, it describes 
 an a?r A B. If the point moves till 
 it comes back through the same point, 
 it has described the whole circle. 
 
 This, as an arc, is called the circumferefice. 
 
 Arcs are said to be equal if they can be made to 
 coincide. Equal arcs may lie either on the same 
 circle or on equal circles. 
 
 § 227. As the arc, like the segment of a line, is 
 described by the motion of a point, it has a sense. 
 
 In future, unless othenvise stated, the sense will 
 be taken as positive if the describing point appears 
 when seen from the centre, or from any other point 
 within the circle, to move from right to left, as in- 
 dicated by the arrow in fig. 112. 
 
 This determines the arc AB. Otherwise we 
 should not know whether the arc A CB or A D B 
 was meant by A B. 
 
 § 228. Sum of Arcs of a Circle. — Between arcs of 
 a circle there exist relations analogous to those betAveen 
 segments on a line. Thus we have 
 AC+CB=AB, 
 AB-\-BA= circumference. 
 
156 Elements of Plane Geometry. 
 
 But if we consider only the amount of motion re- 
 quired to come from A to A^ we may say this is zero. 
 Just as, in the case of an angle, we could add an angle 
 of rotation any number of times to any given angle 
 without altering the figure, so we may now add to any 
 arc a circumference any number of times, without 
 changing the end points of the arc. But we shall 
 understand by the arc A B, unless otherwise stated, 
 the smallest arc described by a point moving in the 
 positive sense from A to B. The arc A A gives, then, 
 either the circumference or the arc zero of no lengtli. 
 Hence we may write 
 
 AB-\-BA=^o, 
 and AB=-BA, 
 
 where now —B A means the arc described by a point 
 
 moved along the circle 
 from ^ to ^ in the 
 
 Fig 
 
 The same equa- 
 tions which hold for 
 .^ segments (§§ 75-82) 
 and for angles (§§ 90- 
 95) hold also for arcs of circles. For instance 
 
 AB + BC-\-CA=o, 
 AB + BC+Cn=AD. 
 
 § 229. Angle at the Centre. — If a point describes 
 the arc AB, the radius drawn to it will describe an 
 angle having its vertex at the centre. This angle is 
 called an a?igleat the centre^ and is said to be subtended 
 by the arc AB, or to stand upon the arc AB. 
 
 § 230. Theorem : Egual arcs subtend equal angles 
 
Angles at the Centre. 157 
 
 at the centre; and conversely^ equal angles at the centre 
 are subtended by equal arcs. 
 
 For if we have the arcs A B and CD equal, we 
 may shde the arc CD^ together with the radii -S C and 
 SD^ along the circle till C coincides with A ; then D 
 will coincide with B^ as CD-=AB. Therefore the 
 angle CSD will coincide with A SB and will be 
 equal to it in magnitude and sense. 
 
 It follows that if A, B, C. . . denote points on the 
 circle, and a, b,c... the radii drawn to these points, 
 then every equation between arcs AB^ B C, &c., will 
 carry with it an equation between the corresponding 
 angles ab, be, &c., and vice versa. 
 
 lfAB=CB>, then a b=cd; 
 
 and also if ab=cd, then AB=CD. 
 
 In the same manner the equation between arcs 
 
 AB+BC=AC 
 
 involves also the equation between angles 
 
 ab + bc=ac, 
 and so on. 
 
 § 231. A diameter, being an axis of symmetry, 
 divides the circle into two equal arcs, called sem/circtes. 
 
 A semicircle subtends an angle of continuation ; or, 
 ez'ery diameter as an angle of continuation is subtended 
 by a semicircle. 
 
 Similarly every right angle at the centre is sub- 
 tended by half a semicircle, called a quadrant of the 
 circle. 
 
 § 232. From the fact that a diameter is an axis 
 of symmetry, other properties follow. 
 
 To every point C on the circle corresponds, with 
 
158 Elements of Plane Geometry. 
 
 regard to a diameter d as axis of symmetry, another 
 p^ ^^ point C on the circle. The line 
 
 CC joining these is therefore per- 
 pendicular to ^ and bisected by it. 
 If we want to find for any 
 point C its corresponding point 
 C with regard to d as axis, we 
 have to draw from C a perpendi- 
 cular to d, and to produce it till it cuts the circle 
 again at C. Then C is the required point, for 
 the point corresponding to a point on the circle lies 
 itself on the circle. This implies the following 
 
 Theorem : A diameter bisects all chords perpendi- 
 cular to it. 
 
 Converse Theorem : If a diameter or a radius bisects 
 a chord which is not itself a diameter, it is perpendi- 
 cular thereto. For. there is but one diameter that 
 bisects a given chord. 
 
 Corollary : The locus of mid points of parallel 
 chords is a diaineter perpendicidar to those chords. 
 
 § 233. If, as in § 232, C, C are points correspond- 
 ing with regard to a diameter ^/(fig. 114), then the 
 arcs CA and A C will be corresponding arcs, and 
 hence equal. The arc CC , and for the same reason the 
 arc CC, is therefore bisected by the diameter d. 
 
 As CA and A C are equal arcs, the angles at the 
 centre subtended by them are equal ; hence angle 
 CSC is also bisected by d. This proves that 
 
 Theorem : The diameter perpendicular to a chord 
 bisects that chord, bisects the two arcs into which this 
 chord divides the circle, and bisects the angles at thg 
 centre subtended by these arcs. 
 
The Tangent. 159 
 
 § 234. We have seen that if a diameter AB is 
 taken as axis of symmetry, then to every point C on 
 the circle corresponds another ^^^ 
 
 point C, such that C C is perpendi- 
 cular to A B. This supposes that 
 the point C does not lie on the 
 axis of symmetry or on the diameter 
 AB. Hence if we draw through 
 one of the end points A of the 
 diameter, a perpendicular to it, 
 this will correspond to itself, and thus can have only 
 the one point A in common with the circle. For if it 
 met the circle again at a point Z>, the point D\ spn- 
 metrical to D -with regard to the diameter AB^ would 
 also he on the circle, or the perpendicular would 
 have three points on the circle, which is impossible 
 (§ 221). 
 
 Definition of a Tangent. — A line which has only 
 one point in common luith a circle is called a tangent to 
 the circle, and that point is called the point of contact. 
 
 The above reasoning gives now the following 
 theorem. 
 
 Theorem : A line through the end point of any dia- 
 meter and perpendicular to it is a tangent to the circle, 
 and has that end point as its point of contact. 
 
 § 235. A tangent may also be considered from 
 another point of view. If we suppose the chord CC , 
 joining symmetrical points, to move away from the 
 centre towards A, then the two end points C and C 
 will approach A from opposite sides, and will at last 
 coincide at A. The line C C , which remains always 
 perpendicular to the diameter AB, will now be a 
 tangent at A. From this we see that a tangent to a 
 
i6o Elements of Plane Geometry. 
 
 circle may be considered as a line which cuts a circle 
 in two coincident points. This also may be taken as a 
 definition of a tangent. 
 
 Second Definition of a Tangent. — If a secant which 
 cuts a circle in two points be moved in any manner till 
 the two points of intersection with the circle coincide^ it is 
 in this last position called a tangent to the circle. 
 
 § 236. At every point A on the circle we can draw 
 a tangent, for we need only draw the radius to A and 
 erect a perpendicular at its end point A. This will 
 be a tangent at A. 
 
 But only one tangent at a point ^ on a circle can 
 be drawn. For if we draw through the point A any 
 other line p not perpendicular to the 
 radius A S^ then the perpendicular 
 from the centre .S to this line will 
 cut it at a point E different from A. 
 Hence the line / must cut the 
 circle again at some other point A' 
 which is symmetrical to A with 
 regard to the diameter SE as axis of symmetr)-. 
 Hence 
 
 Theorem : At every point on the circle one, and only 
 one, tangent can be dratvn to the circle. 
 § 237. This shows also that 
 Theorem : The perpendicular from the centre to a 
 tangent of a circle passes through the point of contact. 
 
 The radius to the point of contact of a tangent is 
 perpendicular to the tangent. 
 
 Both propositions follow from tlie fact tliat but 
 one tangent can be drawn having a given point on 
 the circle as point of contact ; and that there exists 
 but one perpendicular from the centre to a line. 
 
Equal Arcs. 
 
 i6i 
 
 
 Fig. I 
 
 C7- 
 
 1 . . 
 
 C 
 
 ; 
 
 ^ 
 
 \ 
 \ 
 
 \ 
 
 
 1 
 
 \ 
 
 
 J 
 
 3^ 
 
 \ 
 
 ^ 
 
 \B 
 
 § 238. To find other properties let us again con 
 sider a diameter A B 2,"^ axis of 
 s\Tnmetry, and let C C and D D' i 
 be two pairs of corresponding 
 points, then C C and D D' are 
 parallel and the two arcs CD and ^ 
 CD' are corresponding arcs ; 
 hence they are equal. But they 
 are of opposite sense ; or 
 
 CD=-CD\orCD=D'C'. 
 
 In the same manner we have 
 
 CD'= - CD, or CD'=D C, 
 
 where CD' and D C are to be taken in the same 
 sense. 
 
 This is also true if D and D' coincide, so that the 
 secant DD' becomes the tangent at A parallel to the 
 secant C C \ hence 
 
 CA = A C, or CA= - CA. 
 
 This may be expressed thus : 
 
 Th-eorem : i. Two parallel chords cut a circle in 
 points such that the arcs joining one end point of the 
 first to either end point of the second equals the arc join- 
 ing the remaining end points in the opposite sense. 
 
 If CC afid DD' are parallel, then 
 
 CD=D'C and CD'=DC'. 
 
 2. The arcs hetweeii the point of contact of a tangent 
 and the end points of a chord parallel to it are equal 
 and opposite; or 
 
1 62 Elements of Plane Geometry. 
 
 If C C is a chord pa7'allel to the tan gait at A, then 
 CA=AC. 
 
 § 239. The theorems converse to these also hold. 
 
 Converse Theorem \ 1. If two arcs of a circle arc 
 
 equals then the chords which join the i7iitial point of 
 
 either to the end point of the other are parallel. Iti 
 
 sytnbols 
 
 If the arcs CD and D' C ai'e equal, then the chords 
 CC and DD' are pai'allel. 
 
 2. The tangent at the mid point of an arc is parallel 
 to the cho7'd joining the end points of the arc. 
 
 For if arc CZ>=arc D' C , then the diameter ^^, 
 ^ „ which bisects the angle CSC, 
 
 Fig. 118. _ ^ ' 
 
 ^ bisects also the angle DSD', 
 
 '^ ^"^ "--v^ because the angles CSD and 
 D'SC are equal, standing upon 
 equal arcs. Hence C, C and D, D' 
 are pairs of corresponding points ; 
 their joins CC and D D' are 
 therefore parallel. 
 Further, the diameter SA which bisects the arc 
 CC bisects also the angle CSC\ so that C and C are 
 symmetrical with regard to that diameter, and their 
 join is therefore perpendicular to it, hence parallel to 
 the tangent at its end point A, 
 
 § 240. Definition : An angle, luhich has its vertex 
 on the circle and has for limits two chords of the circle, is 
 called an angle at the circumferenxe, j-z/M'Wr^/ by 
 the arc joining the other end points of the chords. 
 
 Thus in fig. 119 the angle A B C '\?> subtended 
 by, or stands upon, the arc A C, both the angle and 
 
 s 
 
Angles at the Circinnference. 
 
 163 
 
 mpplementary 
 
 Fig. 
 
 the arc being taken in the same sense, so that 
 a line through B describes the angle 
 ABC, its intersection with the circle 
 describes the arc A C. 
 
 § 241. If we take in the same 
 circle two angles at the circum- 
 ference with their limits parallel, 
 these will be equal if they have the 
 same sense, as in the first figure, but 
 if they are of opposite 
 
 sense, as in the second _-» c 
 
 figure. In the latter 
 case the angle ABC 
 = angle A'B'C", 
 where B'C is the 
 continuation of CB\ 
 
 If the angles are of 
 the same sense, and therefore equal, they stand upon 
 arcs A C and A'C, which are of the same sense. But 
 as ^^ is parallel to A'B', and BC parallel to 
 B'C, arc ^^' = arc CC, for they are both equal 
 to arc BB\ Hence arc ^<::=arc A'C ; or ^/le 
 angles ABC and A' B'C stand upon equal arcs. 
 
 If the two angles are of opposite sense, .^ ^ is 
 parallel to B'A' and ^C parallel to B' C\ whence, as 
 before, A A' equals CO. 
 
 '^^^tAC=AA' + A'C2indA'C=A'C+CC. 
 
 From this it follows, as before, since AA'=CC', 
 that A'C=AC But angle ABC stands on arc 
 AC, and angle A'B'C stands on an arc equal to 
 a circumference - arc A' C or on arc {-A'C). 
 
 In this case, therefore, the angles ^ ^ C and A' B' C 
 
164 Elements of Plane Geometry. 
 
 stand upon arcs which complete the circumference. 
 Such arcs are therefore said to be supplementary. 
 Hence we have 
 
 Theorem : Two angles at the circiimfei-ence ivhich 
 have their limits parallel stand upon equal arcs if they 
 are equal., but they stand upon supplementary arcs if they 
 are suppletfientary. 
 
 § 242. The converse also holds, as follows : 
 Theorem : If two angles at the circu77ifcrence stand 
 upon equal arcs., and one limit of the one is parallel to 
 one limit of the other., then the second limit to the one is 
 pa7'allel to the second limit of the other and the angles are 
 equal. 
 
 For through the vertex of the second angle only 
 one line can be drawn which is parallel to the second 
 limit of the first angle, and one line only which, to- 
 gether with the first limit of the second angle, makes 
 an angle equal to the first. 
 
 § 243. If we now consider two angles at the 
 ■circumference standing upon the same arc or upon 
 equal arcs, we can slide the one along the circle till 
 its first limit is parallel to the first limit of the second 
 angle; the other limits will then be parallel and the 
 the angles equal. Hence 
 
 Theorem : Two angles at the circumference standing 
 upon the same arc, or upon equal arcs, are equal. 
 Similarly it follows that 
 
 Theorem : Tivo angles at the circumference 7vhich 
 stand upon supplemcjitary arcs are supplementary. 
 And as a special case of the last 
 Corollary : Two angles at the circumference, whose 
 limits meet the circu/nference on the same two points, but 
 
Angles at the Circumfei'ence. 165 
 
 which have their vei'tices on opposite sides of the chord 
 ''oiniiig these points, are supplementary. 
 
 § 244, Considering a tangent as the limiting case 
 of a secant cutting in two coincident points, we 
 get the following theorem — 
 
 Theorem : The angle between the tangent to a circle 
 and a chord which passes through the poi/it of contact 
 is equal to the angle at the circumference standing upon 
 that chord and having its vertex on the side of it 
 opposite to that on which the first angle lies. 
 
 § 245. These theorems may also be stated thus: — 
 
 Theorem : The lines which foiji 
 a?iy point on a circle to two fixed 
 points on it intersect under constant 
 angles. 
 
 § 246. Theorem : Equal arcs, or 
 equal ajigles at the centi-e, or equal 
 angles at the circumference, or sup- 
 plementary angles at the circumference^ 
 a?'e subtended by equal chords. 
 
 And conversely 
 
 Theorem: Equal chords subtend (ist) equal or 
 supplementary arcs; (2nd) equal angles at the cejitre; and 
 (Srd) equal or supplementaiy angles at the circunfe- 
 rence. 
 
 These theorems are proved Hke those in § 230 by 
 sliding the one arc along the circumference till it 
 coincides with the other arc which is equal to it. 
 
 § 247. In the same manner it is proved that 
 
 Theorem : Equal chords are equidistant from the 
 centre. 
 
 And conversely 
 
1 66 Elements of Plane Geometry. 
 
 Fig. 
 
 Chords which are equidistant from the centre are 
 equal. 
 
 § 248. If AB and CD are two unequal chords, 
 AB>Cn, the triangles A SB and CS£>, where 
 ^ denotes the centre of the circle, have the sides 
 SA, SB equal to the sides SC, SD, and of the 
 third sides AB> CD ; there- 
 fore (§ 194) 
 
 /_ASB-> /_csn. 
 
 Of two unequal chords the 
 greater subtends the greater angle 
 at the centre. 
 
 The converse is proved in 
 the same manner. If now the 
 triangle CSD be turned about 
 6^ till SD coincides with SA, then 6" C will fall within 
 the angle A SB at SE^ say. The mid point N oi 
 the chord A E and the centre .S lie thus on opposite 
 sides of AB. The line SN, that is, the perpendi- 
 cular to A E^ will therefore cut AB 3.t some point B. 
 It follows that MS>BS, whilst BS^JfS, if Af is 
 the mid point of AB. Hence the distance iV^^" of 
 the smaller chord A E or CD is greater than the 
 distance SN oi the greater chord AB. Hence 
 
 Theorem : Of t^vo unequal chords that is the greater 
 wJiich is nearer the centre. And conversely 
 
 Of two utiequal chords the greater is nearer the 
 centre. 
 
 Corollary : A diauicier is greater than auy other 
 chord, its distance from the centre being the smallest 
 possible. 
 
Exercises. 1 6y 
 
 Exercises. 
 
 (1) If from points on the circumference of a circle seg- 
 ments of lines be drawn equal, parallel, and of the same sense 
 as a given segment drawn from the centre, what will be the 
 locus of their extremities ? 
 
 (2) Determine the locus of the mid point of a line coimecting 
 a fixed point with a point on the circumference of a given circle, 
 
 (3) One side of a triangle being given, as well as the length 
 of the median line drawn from one of its extremities, find the 
 locus of the vertex opposite the given side. 
 
 (4) A segment of constant length slides with its ends along 
 the limits of a right angle. Find the locus of its mid point. (The 
 carpenter's 'trammel' is an instrument for drawing this locus.) 
 
 (5) The tangents at the extremities of any diameter of a 
 circle are parallel to one another ; and conversely, the straight 
 line which connects the points of contact of two parallel tan- 
 gents is a diameter. 
 
 (6) Each of the angles at the base of the isosceles triangle 
 formed by any chord of a circle and the tangents at its extre- 
 mities is equal to an angle at the circumference subtended by 
 the arc within the triangle. 
 
 (7) Ever}- trapezium inscribed in a circle is symmetrical. (A 
 trapezium is a quadrilateral with two parallel sides.) 
 
 (8) The end points of two equal chords, and likewise those 
 of two parallel chords, of a circle are the vertices of a symmetri- 
 cal trapezium. 
 
 (9) If through any point within or without a circle two 
 lines are drawn cutting the circle, then any one of the four 
 angles contained by them will be equal in magnitude and sense 
 to an angle at the circumference subtended by an arc equal to 
 the sum of the arcs which are intercepted by the limits of that 
 angle produced, if necessary, beyond the vertex. (How ought 
 this theorem to be stated if no account be taken of the sense of 
 the angles ?) 
 
 (10) What corollaries are deducible from the last proposition 
 if the two lines are at right angles to one another ? 
 
 (11) Of all chords which pass through a given point within a 
 circle, which is the shortest? 
 
1 68 Elements of Plane Geometry. 
 
 (12) The length of a chord being given, determine the locus 
 of its mid point. 
 
 DRAWING. 
 
 A circle on paper is generally drawn by aid of a pair of 
 compasses with one point formed by a pencil or a pen. (Take 
 care not to make a big hole with the point inserted at the 
 centre.) For drawing large circles joiners and carpenters often 
 use the ' trammel ' [Ex. (4)]. 
 
 (13) Draw a circle having its centre at a given point and its 
 radius of a given length (take radius equal to, say, \, I, and 2 
 inches). 
 
 (14) At a given point on a circle draw a tangent. 
 
 (15) From a point without a circle draw a tangent to a circle. 
 (This is done by placing the straight-edge through the point 
 and touching the circle.) 
 
 (16) A tangent being drawai to a circle, find the point of 
 contact. (Draw perpendicular from centre.) 
 
 (17) Draw a circle which shall have a given point as centre 
 and touch a given line. 
 
 (18) By aid of set squares divide the circumference of a circle 
 into two, three, four, six, eight, and twelve equal parts. 
 
 (19) In a given circle inscribe regular convex polygons of 3, 
 4, 6, 8, and 12 sides. (A polygon is said to be inscribed in a 
 circle if its vertices lie on the circle. ) 
 
 (20) In a given circle inscribe regular star-polygons of 6, S, 
 and 12 sides. 
 
 (21) Bisect an arc of a circle (by perpendicular from centre 
 to chord). 
 
 (22) Divide the circumference of a circle into 16 equal parts, 
 and draw (in separate figures) all possible regular star-polygons 
 which have the points of division as vertices. 
 
 (23) Divide the circumference of a circle into 24 equal parts, 
 and draw (in separate figiues) all possible regular star-polygons 
 having their vertices at the points of division. How many of 
 these star-polygons are polygons proper, and how many are 
 combinations of regular triangles, squares, &c. ? 
 
169 
 
 CHAPTER XIII. 
 
 THE CIRCLE AS ENVELOPE. 
 
 § 249. Up to this we have only considered the 
 circle as described by a moving point. AVe shall 
 now see that a circle may also be generated by a 
 moving line. 
 
 We need only remember that a tangent has its 
 distance from the centre equal to the radius, and that 
 all lines which have their distances from the centre 
 equal to the radius are tangents, m order to see the 
 truth of the following theorem — 
 
 Theorem : The sd of lines equidistant fj-o?n a fixed- 
 point consists of the tangents to a circle which has the 
 fixed point for its centre and the distance of the lines from 
 it for its radius. 
 
 § 250. If all lines in a set are tangents to a 
 curve, it is said that the lines envelope that curve, 
 or the curve is said to be the envelope of the set. 
 The last theorem thus becomes — 
 
 Theorem : TJie lines equidistant from a fixed point 
 envelope a circle which has the fixed point for its centre 
 and the constant distance for its radius. 
 
 § 251. We have obtained thus a second mode of 
 generating curves. The first was to move a point so 
 as to describe a curve ; the other is to move a line so 
 as to envelope a curve. The first method is con- 
 stantly used in drawing curves by means of a pen 
 or pencil ; the second method is not quite so familiar. 
 
170 Elements of Plane Geometry. 
 
 But fig. 123 will show that a curve may be clearly 
 determined by drawing a number of hues accord- 
 FiG. 123. ing to some law. But we may also 
 
 draw a curve directly by a con- 
 tinuous motion of a line. If we 
 take a plane board, and cover it 
 evenly with a thin layer of some dry 
 powder, such as sand or fine saw- 
 dust, and then move a straight- 
 edge in any manner, always keep- 
 ing its edge on the board, it will 
 remove the sand from those parts 
 of the board swept over by the 
 moving edge, and push it together at other parts. 
 
 These will appear 
 bounded by a curve, 
 and this will be the 
 envelope of the 
 moving line. 
 
 For instance, if 
 a ruler AB be 
 fixed, as in fig. 1 24, 
 to a centre, it will 
 thus trace out a 
 circle, of which, in 
 
 Fig. 124. 
 
 
 ^<^^^ 
 
 
 00^ 
 
 the figure, the greater part is supposed drawn. 
 
 §252. We know (§216) that through one point 
 two lines can be drawn which have the same dis- 
 tance from a given point, provided that that distance 
 is less than the distance between the points. 
 
 Hence if 6" is the centre of a circle, and A a point 
 without a circle— that is, at a distance from the centre 
 
Tangents from a given Point. lyi 
 
 greater than the radius — then two lines may be drawn 
 from A which have their distance from ^ equal to 
 the radius, and which are there- 
 fore tangents to the circle. But if 
 A is on the circle only one such 
 line may be drawn, and if A is 
 within the circle, none. Hence 
 
 Theorem : To a circle two tan- 
 gents 7?iay be di'awji fro7n a point 
 without it, onefro7n a poifit on it, 
 and 7ione fro77i a poi7it withi7i it. 
 
 § 253. The two lines equidis- 
 tant from 6" which can be drawn through a point 
 A are symmetrical with regard to the line SA 
 (§ 216), so are the perpendiculars from .S upon them. 
 Hence 
 
 Theorem : If two fafige/its to a ci7rle be draw7i f'oin 
 a poi7it A without the circle, the7i these are symmetrical to 
 the li7ie A Sjoini7tg A to the ce7it7-e 
 of the circle. These ta7ige7its, together 
 with the 7'adii draw7i to the points 
 of co7itact,for77i a kite. 
 
 Corollary I. : The li7ie SA 
 bisects 
 
 1 St. The angle between the two 
 ta7ige7its. 
 
 2 nd. The angle betwee7i the 7'adii 
 draw7i to the points of contact. 
 
 3rd. The chord joi7ii7ig thepoi7its 
 of co7itact. This is called the 
 CHORD OF CONTACT, a7id is pe7pe7idiciilar to the 
 li7ie A S. 
 
172 Elements of Plane Geometry. 
 
 Fig. 127. 
 
 4th. The ai'cs into which the points of contact divide 
 the circtimference. 
 
 Corollary II.: The diameter perpendicular to a chord 
 passes through the intersection of the tangents drawn at 
 the end points of the chord. For there is but one 
 perpendicular bisector of the chord, and but one 
 hne joining the centre to the join of the two tan- 
 gents. 
 
 Corollary III. : Of the two tangents draivn from a 
 point zvithont a circle tJie segments between the common 
 point and the points of contact aj-e equal in length and 
 equally inclined to the ch ird of contact. 
 
 § 254. Theorem : If A andB are two points without 
 a circle and equidistant from the 
 centre^ then the tangents drawn 
 from A to the circle are equal to 
 those draivn from B, and inchide 
 equal angles ; their chords of contact^ 
 and also the arcs subtended by these 
 chords., are equal. 
 
 For we may turn the circle, 
 together with the tangents drawn 
 from A^ round the centre till A 
 coincides with B^ in which case 
 also the tangents will coincide 
 (fig. 127). 
 
 § 255. Theorem : If A and B (fig. 128) are two 
 points without the circle on the safne diameter , then the 
 tangents from A and B will form a kite, 7C>ith the 
 diameter A B S as axis of symmetry. 
 
 The chords of contact C C and D D' are pa- 
 rallel. 
 
Tangents from a given Point. 173 
 
 Fig 
 
 If A is at a greater distance from S than B, 
 if AS >BS, then the tangents from A are greater 
 than those fvm B, but they include a smaller 
 angle. 
 
 For if we turn B, together with 
 its tangents, about the centre S till 
 B comes to B', a point on the 
 tangent from A, then SB' <SA. 
 We have now from 6" two obliques 
 SB' and SA, of which the first 
 is the smaller. Its departure is 
 therefore the smaller — that is, if 
 D is the point of contact of the 
 tangent from A, DB'<DA. 
 But nB'=CB, so that BC<AD. 
 
 The last part of the theorem 
 may be stated more generally. If 
 we turn B witH the tangents from 
 it to any other position, the length 
 of a tangent drawn from it to the circle remains un- 
 altered. 
 
 § 256. When a circle is given with centre at 6" and 
 a point A without it, it is easy to find, by construction, 
 the points of contact of the tangents drawn from A 
 to the circle. These being known, the tangents ar-: 
 of course found at once, by joining the points of 
 contact to A. 
 
 To find the points of contact, let us draw a circle 
 with centre 6* and radius SA. Then all points on 
 this circle have their tangents to the given circle equal 
 in length. If the radius SA cut the given circle at 
 Z>, and if the tangent at £> cut the new circle at B' 
 
1 74 Elements of Plane Geometry. 
 
 and C, then the chord B' C is perpendicular to 
 Fig. 129. t^e radius. The radius 
 
 therefore bisects the 
 chord, and also the arc 
 B'C, so that arc B'A = 
 arc A C. If we now turn 
 the triangle ^' C" 6" about 
 S, then the chord B'C 
 remains always at the 
 distance SD from S, 
 and remains therefore a 
 tangent to the given circle. If we turn till C comes to A^ 
 then A will come to B' and D to B, where the line 
 SB' cuts the given circle. The line CD takes, 
 therefore, the position AB, touching at B. Hence B 
 is the point of contact of a tangent from A to the 
 circle. The point C where the line SC cuts the 
 given circle is, for the same reason, the other point 
 of contact. Hence we have the following construc- 
 tion : 
 
 Let tJie ti?ie S A cut the given circle at D. Dra^u 
 the tangent at D and let B\ C be the points where this 
 tangent is cut by a circle havi?ig its centre at S and pass- 
 ing throngh A ; then the poi?its B and C 7uhere the lines 
 S B' and S C cut the given ci?'cle are the points oj 
 contact of Um gents drawn from A. 
 
 Exercises. 
 
 (1) Determine the envelope of equal chords of a circle. 
 
 (2) Through a fixed point a perpendicular is drawn to a 
 movable tangent of a circle, and through the mid point of this 
 
Exercises. i 
 
 / D 
 
 perpendicular a line parallel to the tangent. What is the 
 envelope of this line ? 
 
 (3) Two tangents to a circle intersect at constant angles. 
 Find the envelope of the bisectors of those angles. 
 
 {4) The diameters of a circle which pass through the points 
 where two fixed tangents are cut by a movable tangent include 
 constant angles. 
 
 (5) Two parallels are cut by a common perpendicular in A 
 and B, and by another line in P and Q^ in such a manner that 
 the perpendicular OR drawn from the mid point oi A B \o the 
 line PQ divides the latter into two segments PR and RQ^ 
 which are equal respectively to A P and B Q. Find the enve- 
 lope oi PQ. 
 
 (6) A right angle turns about its vertex, which is fixed half- 
 way between two parallels. What is the envelope of the line 
 which joins the points where the limits of the right angle cut 
 the parallels? 
 
 (7) An angle which is equal in magnitude to half the angle 
 between the perpendiculars drawn from its vertex to two given 
 lines turns about its vertex which is equidistant from the given 
 lines. Find the envelope of the line which joins the points 
 where the limits of the angle cut the given lines. 
 
 (8) A kite AB C D, with A B as axis, has the vertex A 
 at a given point and the side CB in a given line. Determine 
 the envelope of the side ^Z). 
 
 BRA IVING. 
 
 (9) Draw figures to the exercises (i)-(5). 
 
 (10) Draw a sufficient number of the set of lines described in 
 questions (6), (7), (8), to bring out the envelope. 
 
 (11) About given circles circumscribe regular convex polygons 
 of 3, 4, 6, 8, and 12 sides. (A polygon is said to be circum- 
 scribed about a circle if each of its sides touches the circle.) 
 
 (12) About given circles circumscribe regular star-polygons 
 of 6, 8, 12, and 16 sides. 
 
 (13) Draw a set of circles 
 
 I. With equal radii having their centres un a given line or 
 
176 Elements of Plane Geometry. 
 
 on a given circle. (They envelope two lines parallel to the 
 given line or two circles concentric to the given circle.) 
 
 2. Having their centres on a given line and touching another 
 given line. (They envelope a third line, passing through the 
 join of the first two.) 
 
 3. Having their centres on a given circle and all passing 
 through a fixed point within, on, or without the given circle. 
 (They envelope a beautiful curve, which is, however, not of an 
 elementary character. ) 
 
 CHAPTER XIV. 
 
 CONDITIONS WHICH DETERMINE A CIRCLE. 
 
 § 257. Let us now see how many conditions are 
 necessar)^ to determine a circle ; in particular, how 
 many points, and reciprocally how many tangents of a 
 circle, may be assumed arbitrarily. 
 
 If it is required to If it is required to 
 
 draw a circle through a draw a circle touching a 
 
 given line a,, then any 
 point 6" may be taken as 
 centre, and its distance 
 Sa from the line as radius. 
 
 If a circle has to be 
 drawn touching two given 
 lines a and b, the centre 
 has to be equidistant from 
 a and b ; hence it must 
 lie on one of the bisec- 
 tors of the angles ab 
 (§ 2°8). 
 
 given point A, any point 
 6" may be taken as centre, 
 and its distance SA from 
 the point as radius. 
 
 If a circle has to be 
 drawn through two given 
 points A and B, the centre 
 has to be equidistant from 
 A and B ; hence it must lie 
 on the perpendicular bisec- 
 tor of AB (§ 208). 
 
Circle tJiroiigh tJiree Points. 
 
 i-jj 
 
 Theorem 2.'. An infinite 
 nnmber of cii'des may be 
 drawn through two given 
 points ; the locus of their 
 centres is the perpendicular 
 bisector of the segment join- 
 ing the tivo given points. 
 
 § 258. A circle, then, is not determined by two 
 points or by two tangents. But 
 
 Theorem b : An infinite 
 number of circles may be 
 draion which touch two 
 given lines; the locus of 
 their centres consists of the 
 bisectors of the angles between 
 the given lines. 
 
 if three points A, B, C are 
 given, we know (§ 210) 
 there is always one, and 
 only one, point .S equidis- 
 tant from A,B, C, provided 
 that these points do not 
 lie in a line. The circle, 
 having this point 6" as 
 centre, and SA=SB 
 =^SC as radius, will 
 pass through A, B, and C, 
 and this will be the only 
 circle through them. It is 
 said to be circumscribed 
 
 if three lines (Z, ^, c are 
 given, we know (§212) there 
 are always four points 
 .S, Si, S2, 6*3 equidistant 
 from the three lines a, b, 
 and c, provided that the 
 three lines form a triangle 
 Each of these is the 
 centre of a circle touching 
 all three lines. One has its 
 centre within the triangle 
 a be, and is said to be 
 inscribed in it ; the others 
 are said to be escribed. 
 
 about the triangle ABC. 
 
 We thus see that three points determine one 
 circle, whilst by three tangents four circles are deter- 
 mined. The first case is the more important one. It 
 gives the following theorem : 
 
 Theorem : Through three points not in a line, one. 
 
 and only one, 
 thint^ 
 
 circle can be drawn \ or, what is the same 
 
 N 
 
178 Elements of Plane Geometry. 
 
 Fig. 130. 
 
 Two circles which have three points in common 
 coincide. 
 
 § 259. The contra-positive form of the last propo- 
 sition gives 
 
 Theorem : Two different circles cannot have more 
 than two points in conunon. 
 
 Two circles, however, need 
 not have any point in common. 
 For instance, if we call two circles 
 which have the same centre concen- 
 tric (fig. 130), we see at once that 
 Two concentric circles with 
 unequal radii have no point in 
 common; the one which has the 
 smaller radius lies altogether luithin the other. 
 
 If two circles have two points in common they are 
 said to cut, or to intersect, and the line joining the 
 common points is called the common chord. 
 
 It may also happen, as we shall presently see, that 
 two circles have only one point in common. In this 
 case the two circles are said touchy and the common 
 point is called the point of contact. 
 
 § 260. To investigate these different cases let us 
 suppose any two circles given, with centres at different 
 
 points S and S'. The 
 line S S\ called the line 
 of centres, contains a 
 diameter of each, and 
 is therefore a common 
 axis of symmetry. 
 
 Theorem: 7 wo circles 
 have always a common axis of symmetry, and in general 
 only one, viz. their line of centres. If they have two 
 
 Fig. 
 
Ti\:o Circles. 
 
 1/9 
 
 co7nmon axes of symmetry^ and are unequal, they arc 
 C07icentric and have an infaiite niwiber of such axes. For 
 in this case the line of centres becomes indeterminate. 
 
 § 261. From this there follows that 
 
 If two circles have one poi?2t A in common which is 
 not on the line of centres, then they have necessarily a 
 second point A' in common, viz. the point which is, 
 with regard to the line of centres, symmetrical to A. 
 
 If two circles have two 
 points in common, these 
 must be corresponding 
 points with regard to the 
 
 Fig. n2 
 
 line of centres, for there 
 are but two points of 
 intersection possible, and 
 to a point of intersection 
 corresponds a point of intersection. Hence 
 
 Theorem : The common chord of two intej'secting 
 circles is bisected perpendicularly by the line of centres. 
 
 § 262. Contact of Circles. — If, however, the two 
 circles have a point A in common, which lies on the 
 line of centres, then they cannot have any other point 
 in common, as the point A corresponds to itself; and 
 conversely, if two circles ^^ 
 
 have bu t one point in com- 
 mon, this point must lie 
 on the line of centres. Or 
 
 Theorem : If two 
 circles touch, the point of 
 contact lies on the line of 
 centres, and the line per- 
 pendicular to the latter through the point of contact is 
 a common tangent. 
 
i8o Elements of Plane Geometry. 
 
 § 263. Different Positions of two Circles.— If two 
 circles are given they may have one of the following 
 positions : 
 
 If the two circles have two different points A and 
 B in common, the two circles intersect. A point 
 which moves along one of the circles, crosses the 
 other circle on passing through A^ and recrosses on 
 passing through B. Hence if the moving point is 
 first without the second circle, it will lie within it when 
 it has passed A, It will remain ^\^thin it till it comes 
 to B^ and will be outside after having passed B. Two 
 circles,therefore, which intersect, lie each partly within 
 and partly without the other. If, however, two circles 
 touch, then a point on moving along the one circle 
 will never cross the other circle ; hence it will always 
 remain within or without the second circle, according 
 as it was at first Avithin or without it. 
 
 Of two cif'cles which touch, the one lies either 
 altogether z^ithin the other, or each of them lies altogether 
 without the other. 
 
 The circles are said to have in the first case 
 internal, in the second exterjial, contact. If two circles 
 have no point in common, one lies either totally within 
 the other or each lies without the other. 
 
 § 264. These relations will become still clearer 
 from the following considerations, which at the same 
 time will enable us to state the conditions wliich must 
 hold in each case between- the radii of the circles and 
 the distance between the centres. 
 
 Let 5i and S^ be the centres of the circles ; let r, 
 and ro denote their radii, ;'2 beingsupposedthe smaller ; 
 and let d denote the distance between the centres. 
 
Positions of tzuo Circles. 
 
 Fig. 
 
 hence 
 
 Fig. 13= 
 
 ^Ve shall at first suppose that the circles are con- 
 centric, and then that the centre S2 of the smaller circle 
 moves along a line to a greater and greater distance 
 from Si. 
 
 If the circles are concentric the smaller one, with 
 radius r^, will lie altogether within the other. If 
 we next move ^2 to the position in 
 fig. 1 34, we see that S^ S<i together 
 ^^'ith S2 A is less than S^ B, or that 
 SiB — S^AySiS^, or ri—i\>d\ 
 hence also r^ + 1\ > d. This goes on 
 till A coincides with B (fig. 135) : in 
 this case the circles touch internally. 
 Now S^S^-^- S^A^ 5i B, or i\ — r, 
 ri^r,>d. 
 
 If ^"2 moves further, the two 
 circles will cut at two points C and 
 D. S^CSo is a triangle (fig. 136). 
 Hence the difference of two sides 
 Si C and .So C is less, and their 
 sum greater than the third side SiSo : 
 or r^—r^Kd, ri+r2>d. 
 
 Moving S2 still further, the circles come to a 
 position of external contact. In 
 this case (fig. 137) the distance 
 between the centres is equal 
 to the sum of the two radii, or 
 ^1 -f 7'.2 = ^' Hence also 
 ri—r2<d. 
 
 On increasing d again, the 
 circles will separate and have no point in common 
 (fig. 138). Now the distance d is greater than the 
 
l82 
 
 Elements of Plane Geometry. 
 
 sum of the two radii. Hence i\-\-i\<d. We may 
 therefore classify the different possible cases thus : 
 
 I. One circle lies ivithin the other, r^—r.2>d. 
 
 II. I7ite7'7ial co7itact, r^—r2^=d. 
 
 III. Litersection, ;'i — r^Kd ; r, + r.j > d. 
 
 IV. Exte7-nal contact, r^Arr^j^d. 
 V. One circle lies e7itirely without the other, r^ + r^ < d. 
 
 Fig. 137. 
 
 Fig. 138. 
 
 § 265. If now two circles are given, then one of 
 the above five positions must hold, and only one can 
 hold. Of the above five relations also one, and only 
 one, must hold. For the difference r,— ramustbe 
 less than, equal to, or greater than d; and likewise 
 the sum r^ + ^2 ri^^^st be either less than, equal to, or 
 greater than d. 
 
 Combining these results, we get the above five 
 cases, and these only. For in the first two cases the 
 difference between the radii is greater than, or equal 
 to, the distance between the centres ; hence the sum 
 of the radii is necessarily greater than that distance. 
 Similarly in the last two cases the sum of the radii is 
 equal to or less than the distance between the centres ; 
 the difference between the radii is therefore certainly 
 less than that distance d. 
 
 From this it follows at once that if we knoiv the 
 
Couiinoii Tangents of two Circles. 
 
 i8 
 
 relations between the difference and the sum of the radii 
 as compared to the distance beticeen the centres, then we 
 also knoiu in luhich of the five possible positions the tivo 
 centres lie. 
 
 If, for instance, we know that the sum of the radii 
 equals the distance between the centres, then we know 
 that the circles must have external contact, because 
 this is the only case in which the relation in question 
 
 holds. 
 
 The same reasoning applies to the other cases. 
 
 § 266. Common Tangents to two Circles.— All 
 these last results (from § 259 to § 265) depend upon 
 the property that two circles passing through the 
 same three points coincide. Theorems reciprocal to 
 these do not exist, because three tangents determine 
 four circles ; hence two circles which touch the same 
 three lines do not necessarily coincide. 
 
 From this it follows that two circles may have 
 at least three common tangents. But two circles have 
 their line of centres as an axis of symmetry. To a 
 common tangent corresponds, with regard to that 
 axis, a common tangent to the two circles. 
 
 It follows, therefore, that the tangents common to 
 two circles occur in pairs, corresponding ones inter- 
 secting on the axis of symmetry. Hence as our circles 
 have three common tangents, they must in general 
 have four tangents in common. This is the greatest 
 number of tangents which two circles can have m 
 common. 
 
 A fuller investigation of the tangents common to 
 two circles may be postponed till after the investiga- 
 tion of similar figures. 
 
184 Elements of Plane Geojnetry. 
 
 § 267. Four Points on a Circle. — As three points 
 determine a circle, it is in general impossible to draw 
 a circle which passes through four given points 
 A, B, C,D ) the circle determined by the three points 
 A, B, C will, as a rule, not pass through the point D. 
 But it is possible that this may happen. In this case we 
 have four points on a circle. Such 
 points are said to be concydic. 
 
 If we join two of these — say, A 
 and B — to each of the others, we 
 obtain two angles at the circum- 
 ference A CB and ADB. These 
 stand either on the same side 
 (fig. 139) or on opposite sides of AB (fig. 140), and 
 hence either on the same or on sup- 
 plementary arcs of the circle. In 
 the first case the angles are equal, in 
 the second they are supplementary. 
 Theorem : If four points are 
 concydic, then the angles included by 
 the lines which join two of the points 
 to each of the others are either equal 
 or supplementary, according as they lie on the same or on 
 opposite sides of the line Joining the first two points. 
 
 § 268. duadrilaterals Circumscribed about a 
 Circle. — A quadrilateral whose sides are tangents to 
 a circle is said to be circumscribed about a circle. 
 
 As three tangents determine a finite number, 
 namely four circles, it is, as a rule, impossible to draw 
 a circle which shall touch four given lines, or shall be 
 inscribed in a quadrilateral. If a quadrilateral be 
 such that a circle may be inscribed in it, it nuist have 
 
CirciLinscrihed Ojiadrilateral. 
 
 185 
 
 properties reciprocal to those considered in the last 
 paragraph. To bring out this reciprocity we will state 
 the above theorem about a quadrilateral inscrib ed in 
 a circle in a slightly different form, together with the 
 reciprocal theorem, as follows : — 
 
 If a quadrilateral he 
 inscj'ibed in a circle^ then 
 either the siuns, or else the 
 differences of paij's of oppo- 
 site angles, are equals 
 according as the quadi'ila- 
 tcral is convex or not. 
 
 If a quadrilateral be 
 circumscribed about a circle, 
 then either the su?ns or 
 else the differences of op- 
 posite sides are equal. 
 
 The sums are equal if 
 the quadrilateral is convex, 
 or has a re-entrant angle, 
 but the differences are equal 
 if two opposite sides inter- 
 sect. 
 
 ^-''Q"^^ 
 
 d 
 
1 86 Elements of Plane Geometry. 
 
 In fig. 141 a, b, c,d are four tangents to a circle ; 
 they determine three different quadrilaterals according 
 as ^ or ^ or ^ be taken as the side opposite a. 
 
 In the quadrilaterals PQPQ and PRPR it 
 is the sum of opposite sides which is constant, whilst 
 in the quadrilateral QR QR', of which the opposite 
 sides, QR', QR, intersect at P, it is the difference. 
 
 The proof of this theorem follows at once from 
 the fact that the tangents drawn from a point to a 
 circle are equal in length (§ 253). 
 
 To take the first quadrilateral, we have (fig. 141) 
 
 PA = PB, QA = QD, QC= QB, PC^P'D. 
 
 Hence PA -^ AQ+ PC + CQ = PB + BQ 
 + P'D + DQ- 
 
 or, PQ + PQ=PQ: + PQ. 
 
 And this is the theorem. The other cases are 
 proved in the same way. 
 
 Exercises. 
 (i) If a point has equal distances from more than two points 
 on a circle, it is the centre of that circle. 
 
 (2) State the theorem contra-positive to the last, and discuss 
 the question vi'hether two equal segments may be drawn from 
 any point to points on a circle. 
 
 (3) Two intersecting circles make equal angles at the two 
 points of intersection. 
 
 (By the angle between two intersecting circles is meant the 
 angle between the tangents to the circles at the point of inter- 
 section. The circles are said to cut orthogonally when this angle 
 is a right angle.) 
 
 (4) What is the magnitude of the angle made by two circles 
 each of which passes through the centre of the other ? 
 
 (5) If two circles cut one another orthogonally, the two 
 
Exercises. 187 
 
 radii to each point of intersection will be perpendicular to one 
 another. Prove also the converse of this, and examine in which 
 case the common chord will be equal to the segment between 
 the two centres. 
 
 (6) A circle may be circumscribed about ever}' symmetrical 
 trapezium and inscribed in every kite. 
 
 (7) In which case may a kite be inscribed in, or a symmetri- 
 cal trapezium be circumscribed about, a circle? 
 
 (8) A parallelogram inscribed in a circle is a rectangle. 
 
 (9) A parallelogi-am circumscribed about a circle is a 
 rhombus. 
 
 (10) Two equal circles have the mid point between their 
 centres a centre of symmetiy ; they have also two axes of s)Tn- 
 metry. If the circles touch, the point of contact is the centre of 
 symmetry. 
 
 (11) How has a line through one of the common points of 
 two intersecting circles to be drawoi in order that the two circles 
 may intercept equal chords on it ? 
 
 (12) Through one of the points of intersection of two circles 
 draw the line on which the two circles intercept the gi-eatest 
 possible segment. 
 
 (13) The centres of two circles, whose radii are five and six 
 inches respectively, are four inches apart. Will the circles 
 intersect or not ? How far apart must their centres be placed 
 in order that the circles may touch? 
 
 (14) If any two lines be dra\^Ti through the point of contact 
 of two circles, the joins of their second intersections with each 
 circle wall be parallel to one another. 
 
 DRA WING. 
 By the aid of a pair of compasses perform the following 
 five constructions: 
 
 {15) Draw a triangle which has its sides of given length. 
 
 (16) On a given base constract an isosceles triangle. 
 
 (17) Bisect a given segment. (Construct a kite having the 
 segment as its transverse axis.) 
 
 (iS) Bisect a given angle. (Construct a kite having the given 
 angle at one end of the axis.) 
 
1 88 Elements of Plane Geometry. 
 
 (19) Through a given point on, or off, a given line draw a 
 perpendicular to that line. 
 
 (20) Draw a circle 
 
 1. Through three given points (circumscribed about a 
 triangle). 
 
 2. Inscribed in a triangle. 
 
 3. Having a given radius and touching two given lines. 
 
 4. Having a given radius passing through a given point and 
 touching a given line or a given circle. 
 
 5. Touching a given circle at a given point and passing 
 through a given point. 
 
 (21) Draw figures to the above exercises, from (3) to''(i4) in- 
 clusive. '^ " 
 
 LONDON : PKINTED BY 
 
 SPOTTISWOODE AND CO., NEW-STREET SQUARE 
 
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