logical foundations of geometry

7/31/2019 Logical Foundations of Geometry

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Logical Foundations of

Geometry

Prof. P. C. Joseph

Revised and Edited by

Prof. Sebastian Vattamattam

Cover designed byK. R. Suresh


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Copyright 2011 Ms. Rosemary M. George

All rights reserved

Acknowledgement

Mrs. Rosamma Joseph


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Prof. P. C. Joseph(1909 - 1973)

Professor P. C. Joseph was born on 18 February 1909. His school educa-tion was in St. Berchmans English School, and he passed Intermediatefrom St. Berchmans College, Changanassery. In 1930 he took B. A.(Hons) in Mathematics from Maharajas College of Science, Trivandrum.Then he joined the Mathematics Department of St. Berchmans College,where he continued teaching for sixteen years. During this period, hetook M. A. (Hons) in Mathematics from Madras University.

In 1946 Prof. Joseph left Kerala to join St. Philominas Science Col-lege, Mysore. There he worked for four years as Assistant Professor andHead of the Department of Mathematics. In 1950 he returned to Keralaand joined St. Thomas College, Palai as the first Professor and Head ofthe Department of Mathematics. He continued in this position for twodecades until his retirement in 1970.

Throughout his teaching career, Prof. Joseph was immersed in ex-ploring new vistas of knowledge. In 1961 he went to Calcutta to attenda Summer Course (Advanced) in Statistics, at Indian Statistical Insti-tute. One of his research papers was published in the prestigious journal,American Mathematical Monthly.1

Prof. Joseph was one of the founding members of Kerala Mathemat-ical Association. Though Mathematics was very close lo his heart, Prof.

Joseph had an unusual fascination for learning languages. Let me quoteRev. Fr. Thomas William, former principal of St Berchmans College,

Mr. P. C. Joseph has impressed me as a man of versatile tal-ents. He is a linguist of the first order and knows Malayalam,

1A Generalization of Hjemslevs Theorem, American Mathematical Monthly,Vol.74, No.5, May 1967


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Kannada, English, French, German, Sanskrit, and Hindi. Inthe last one, he has taken the Rashtra-Bhasha-Visarad De-gree. He is a good speaker in Malayalam and English. Be-sides the various branches of Mathematics, he has acquiredproficiency also in Politics and Economics.

All this intellectual achievement has for its background a veryunassuming personality. Mr. P. C. Joseph is a very simpleoutspoken gentleman.2

2Extract from a letter dated 24 March 1948


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Preface

Professor P. C. Joseph was my Guru. I was in a batch of about sixteenstudents he taught in the last four years of his career. His only daughter,Rosemary, too was one among us. Our beloved Professor died on 21 De-cember 1973, leaving behind a couple of unpublished manuscripts. Theywere handed over to me by his wife, Mrs. Rosamma Joseph. This bookis the edited and completed version of one of those manuscripts.

This book introduces all the fundamental principles of Euclidean Ge-ometry, at the school level, in a comprehensive, logical way.

Working to bring out this book, is my gurupooja.

Sebastian Vattamattam


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Glossary

subset ofintersection

belongs to/ does not belong tod(A, B) or dist AB distance between A and B

AB or disp AB displacement from A to B|x| absolute value of x< less than> greater thanseg AB line segment ABray AB ray AB

union triangle

A angle AmA measure of angle Ast straight angle= congruent to or if f if and only if corresponds to= not equal to parallel to

perpendicular to

similar toar ABC area of ABCR set of real numbersA B Cartesian product of A and B empty setf : A B function f from A to BG(f) graph of f

or

implies

or implies and is implied by p or p not p equivalent to

andor


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Contents

1 Points and Lines 1

1.1 Elements of Set Theory . . . . . . . . . . . . . . . . . . . 1

1.2 Points and Lines . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Incidence Relations . . . . . . . . . . . . . . . . . . . . . 3

1.4 Distance and Ruler Axiom . . . . . . . . . . . . . . . . . . 51.5 Choosing a Coordinate System . . . . . . . . . . . . . . . 8

1.6 Rays or Half-lines . . . . . . . . . . . . . . . . . . . . . . . 11

1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Convexity, Half-planes and Angles 15

2.1 Convex Set . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Geometric Figures . . . . . . . . . . . . . . . . . . . . . . 172.3 The Protractor Axiom . . . . . . . . . . . . . . . . . . . . 20

2.4 Pencils and Quadrants . . . . . . . . . . . . . . . . . . . . 27

2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Congruent Figures 29

3.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Introducing Congruence Relation . . . . . . . . . . . . . . 30

3.3 Congruence of Triangles . . . . . . . . . . . . . . . . . . . 31

3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Triangles and Parallel Lines 41

4.1 Exterior Angle Theorem. . . . . . . . . . . . . . . . . . . . 41


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CONTENTS xi

11 Coordinate Geometry and Trigonometry 135

11.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . 13511.2 Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

11.3 Trigonometric Ratios . . . . . . . . . . . . . . . . . . . . . 145

11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

12 Mathematical Logic 159

12.1 Logical Statements . . . . . . . . . . . . . . . . . . . . . 159

12.2 Connectives . . . . . . . . . . . . . . . . . . . . . . . . . . 16312.3 Relations Between the Connectives . . . . . . . . . . . . . 165

12.4 Converse, Inverse and Contrapositive . . . . . . . . . . . . 167

12.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

12.6 Requirements for a Valid Inference . . . . . . . . . . . . . 169

12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 170


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xii CONTENTS


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Chapter 1

Points and Lines

1.1 Elements of Set Theory

In Geometry we deal with entities like lines, triangles, and circles eachof which is a set of points. A dot is a model of a point, which has noconcrete existence. In fact a point cannot be defined. We can think ofa set as a collection of objects. But no formal definition can be given toa set.1 Such terms that we accept without formal definition, are calledprimitive terms. We shall start with examples of sets.

Example 1.1.1 A =

{a,b,c

}is the set of elements a,b,c. For a is an

element of A we write a A. x / A means x is not an element of A.

Definition 1.1.1 Empty Set: A set with no element is called an emptyset, denoted by . A set with only one element is called a singleton set.

Definition 1.1.2 Subset: If every element of a set P is an element ofa set Q, then we say, P is a subset of Q and write P Q. In example1.1.1, B =

{a, b

}is a subset of A.

1A set, suppose, can be defined. Then, can a set be its own element ? Whateverbe your answer, let S denote the set of sets none of which is an element of itself.Now, is the set S an element of itself or not ? Whether you answer yes or no, youwill reach a contradiction. This is what is called Russells Paradox, and it shows theimpossibility of defining a set. There is a similar paradox, which is more popular:In a town, there is a barber who shaves all those, and only those, who do not shavethemselves. Now the question is, Does the barber shave himself or not ? Try toanswer this question.


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2 Points and Lines

Definition 1.1.3 Union and Intersection: If A, B are two sets, thenthe set of elements in A or B or in both, is called the union of A anB, denoted by A

B. The set of elements in both A an B is called the

intersection of A an B, denoted by A

B.

Example 1.1.2 If A = {a,b,c} and B = {b,c,d,e}, then AB ={a,b,c,d,e} and AB = {b, c}See the figure 1.1, called Venn diagram, representing A

B and A

B.

Figure 1.1: Union and Intersection

Definition 1.1.4 Disjoint Sets: Two sets A and B are disjoint if theycontain no common element, i.e. if A

B =

Definition 1.1.5 Partition of a Set: Suppose a set X is the unionof a collection of mutually disjoint non-empty subsets of itself. Then thecollection of subsets is called a partition of X.

Example 1.1.3 If X = {a,b,c}, then the collection {{a, b}, {c}} of itssubsets is a partition of X. {{a}, {b}, {c}} is another partition.

1.2 Points and Lines

In this chapter we present some axioms, definitions, and theorems onpoints and lines. It is impossible to define every term using only pre-viously defined terms. Terms which are undefined are called primitiveterms.


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1.3 Incidence Relations 3

We intend to prove every proposition.2 But some of the earlier propo-sitions cannot be proved. They are only assumptions accepted withoutproof. Such assumptions are called axioms. Every proved proposition iscalled a theorem. The proof of a theorem depends on the axioms, defini-tions, and previous theorems.

We choose set, point, line, plane and distance as our primitive terms.

The student will readily accept that a line or a plane is a set of points.

Lines and planes are subsets3

of the set of all points, which is called space.A tightly stretched string and a ray of light are models of a line. A

sheet of paper spread on a drawing board, the ruler, and the sharp tip ofa pencil are models of plane, line, and point respectively.

In what follows, the formal statements of some axioms, definitions,and theorems are given. Many of the theorems are stated without proof.

1.3 Incidence Relations

In this section we list the first group of axioms and a few theorems anddefinitions.

Axiom 1.3.1 Given two distinct points, there is a unique line containing(or joining or passing through) them.

In other words, the line, regarded as a set of points, is uniquely de-termined by two members of the set. This is only a formal statement ofour experience. We can draw only one straight line joining two dots on asheet of paper; there is only one straight path from one place to another;only one beam of light, from a given source A, can illuminate a givenobject B.

Definition 1.3.1 Collinear Points: A set of points is said to be collinear,if there exists a line containing all the points of the set.

Axiom 1.3.2 There exist non-collinear points.

2See Defn. 12.1.13See Defn. 1.1.1


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4 Points and Lines

This is an assertion that all the points of space are not in a singleline.4

Axiom 1.3.3 Given any three non-collinear points, there exists one andonly one plane containing the given points.

In other words, a plane, regarded as a set of points, is uniquely deter-mined by three non-collinear points. A table with only three legs can bein equilibrium even if all its legs are of unequal length. This is because

the feet of the three legs determine a plane. But a four-legged table canbe in equilibrium only if all the legs are of equal length. In other words,the foot of each leg should be in the plane determined by those of theother three legs.

Axiom 1.3.4 If two distinct points are in a plane, then the line contain-ing these points is also in the same plane.

Definition 1.3.2 Coplanar Points: A set of points is said to be copla-nar if there exists a plane containing all the points of the set.

Axiom 1.3.5 There exist non-coplanar points.

This means that all the points of space are not in a single plane.

Axiom 1.3.6 If two distinct planes intersect, then their intersection is

a line.

If walls are taken as models of planes, we observe that two adjacentwalls of a room meet along a line.

The following are some theorems that follow from the above axioms.

Theorem 1.3.1 Two distinct lines do not intersect in more than one

point.

That is, considering two lines as sets of points, their intersection will beeither an empty set or a singleton set.5

4A line, being a set of points, it is only natural to say that a point is in a line.But we are used to say that a point is on a line.

5See definitions 1.1.3 and 1.1.1


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1.4 Distance and Ruler Axiom 5

Theorem 1.3.2 Given a line and a plane, only one of the following ispossible:1. The line lies in the plane.2. The line and the plane intersect at a single point.3. The line and the plane have no point in common.

Mathematical statements can be made more precise by using symbols.It is necessary that students get acquainted with the symbolic languageof Mathematics. With this in mind, we restate the above theorem usingsymbols.6

Restatement. Given a line l and a plane P, only one of the followingis possible:

1. l P.2. l

Pis a singleton set.3. l

P= .Theorem 1.3.3 Given a line and a point not on the line, there is exactlyone plane containing both the line and the point.

Restatement. Given a line l and a point A / l, there is exactly oneplane Psuch that l Pand A P. 7

1.4 Distance and Ruler Axiom

A positive number called distance is associated with every pair of pointsin space. This is stated as the next axiom.

Axiom 1.4.1 To every pair of points P, Q in space, there correspondsexactly one non-negative number, denoted by distPQ or d(P, Q) suchthat

1. dist P Q > 0 if P and Q are distinct

2. distPQ = dist QP, and3. distPQ = 0 iff P and Q are the same.

[distPQ is the distance between the points P and Q]We measure distance between two points by means of a scale or graduated

6Such restatements can be skipped without any loss of continuity.7For the meaning of / see Defn. 1.1.1


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6 Points and Lines

straight edge. The essentials of a scale are a straight edge and marks alongthe edge with numbers denoting the marks such that for any two marksA, B with numbers a,b,

distAB = |b a|.The symbol |x| denotes the absolute value of x, defined by the rule:|x| =

x if x 0,x if x < 0

Thus for marks A and B on the scale denoted by the numbers 5 and1 respectively, dist AB = |1 5| = 4.

Although we can use only scales of finite length, we can think of scalesof infinite lengths also. We often measure distances along a line from achosen point O in either of two opposite directions. To distinguish be-tween the two directions we call one of them the positive direction andthe other the negative direction.

We sum up our experience of measurement and our idea of a scale of

infinite length, in the axiom given below.

Axiom 1.4.2 Birkhoffs Ruler Axiom8 Any given line, regarded asa set of points, can be placed in one-one correspondence9 with the set ofreal numbers such that

1. to every point in the line, there corresponds exactly one real num-ber,

2. to every real number there corresponds exactly one point on theline,

3. a given point O on the line corresponds to 0,

4. another given point A corresponds to a positive number, and

5. the distance between any two points is the absolute value of thedifference of the corresponding numbers.

Definition 1.4.1 Coordinate System: The correspondence describedin the above axiom 1.4.2 is a one-one correspondence and it is called acoordinate system for the line. The number that corresponds to a pointmay be called its coordinate.

8Garrett Birkhoff (1911-1996), American mathematician.9See definition 3.1.4


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1.4 Distance and Ruler Axiom 7

The point O with coordinate 0, in a coordinate system, is called theorigin. The coordinate x of a point P is also called the directed distance

or displacement of P from O, abbreviated as dispOP or OP.10

Definition 1.4.2 Displacement: The directed distance of a point Bfrom a point A is called displacement AB, denoted disp AB.

disp AB = disp OB dispOA = OB OA

Ifa andb are the coordinates of the points A andB respectively, then,

disp AB = b aTheorem 1.4.1 If A,B,C are three collinear points, then, disp AB =disp AC+ dispCB

Proof Let O be the origin of a coordinate system for the line containingthe given points.

disp AC = dispOC dispOA (1.1)

dispCB = dispOB dispOC (1.2)Adding (1.1) and (1.2) we get

disp AC+ dispCB = dispOB dispOA= disp AB

Note. dispAP = dispPA, for x a = (a x)Example 1.4.1 We shall suppose that figure 1.2 satisfies the Ruler Ax-

iom 1.4.2. Then,

dist OP = |x 0|= |x|

dispOP = x 0= x

dist AB = | 2 2.5|

= | 4.5|= 4.5

disp AB = 2 2.5= 4.5

10Note for teachers: The directed line segmentOP is the position vector of P in

the one-dimensional vector space with O as the origin. AlsoOP = x, the coordinate

ofP. The sign ofx determines the direction of the vectorOP.


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8 Points and Lines

Figure 1.2: Distance and Displacement

1.5 Choosing a Coordinate System

Any number of coordinate systems can be chosen for a given line. Butthere is a simple relation between any two of them.

Theorem 1.5.1 Any two coordinate systems for a line are connected bya relation of the form x = x a, where a is a given number, and x, xare the coordinates of the same point in the two systems.

Restatement Let O and O be the origins of two coordinate systems

where the coordinate of O in the first system is a. If P is a point on theline having the coordinate x in the first system, then its coordinate x inthe second system is either x + a or x a.Proof [Figure 1.3] Let b and x be the coordinates of O and P respec-

Figure 1.3: Two Coordinate Systems


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1.5 Choosing a Coordinate System 9

tively, in the second system with origin O. Then,

dispOO = a

dispOP = x

dispOO = b

dispOP = x

x = dispOP

= dispOP dispOO= x

b (1.3)

dispOO = a,

dispOO = b

dist OO = |a|= distOO

= |b|Therefore, b = a (1.4)

From (1.3) and (1.4), we get

x = x + b

= x a

Definition 1.5.1 Betweenness: A point B is said to be between thepoints A and C if A,B,C are distinct collinear points and dist AB +distBC= dist AC

We have two obvious inferences from the Ruler Axiom 1.4.2 and thedefinition of betweenness.

Theorem 1.5.2 Let A,B,C be three collinear points with coordinatesa,b,c, respectively. ThenB is betweenA andC iffa < b < c ora > b > c.11

Theorem 1.5.3 Given three distinct points on a line, one and only oneis between the other two.

Theorem 1.5.4 Let two points A and B be on a line, and P be anyother point on the same line. Then one and only one of the followingstatements is true:

11For the meaning of iff see definition 12.1.2


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10 Points and Lines

1. P is between A and B

2. B is between A and P3. A is between B and P

IfB is between A and P we may say P is beyond B, and ifA is betweenB and P, P is beyond A.

Definition 1.5.2 Open Segment: The set of points between A and Bis called the open segment AB.12 The set formed by adjoining A and

B to this open segment is called the closed segment AB. The distancebetween A and B is called the length of the segment AB, whether it isclosed or open. A point M is called a midpoint of AB if M is betweenA and B and distAM = distMB.

Theorem 1.5.5 Every segment has exactly one midpoint.

Proof Let a and b be the coordinates of A and B, respectively, referred

to some coordinate system. Let M be the point whose coordinate is

a+b

2 .Then, a < a+b2

< b or a > a+b2

> b. Therefore M is between A and B.Further,

distAM =

a + b2 a

=

b a2

distMB =b a + b2

=

b a2

Hence M is a midpoint of AB.

Next, let us show that the point M is unique.

If N is a midpoint having the coordinate n, then,a < n < b or a > n > b and

dist AN = distNB

|n a| = |b n|12The set of real numbers, corresponding to this set of points, is called an open

interval, denoted by (a, b) where a, b are the coordinates ofA,B.


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1.6 Rays or Half-lines 11

This means that n a = b n or n a = n bBut, n

a

= n

b since a

= b

Therefore,

n a = b n (1.5)n =

a + b

2. (1.6)

There is only one point with coordinate a+b2

, by the Ruler Axiom 1.4.2.Hence there is only one midpoint.

1.6 Rays or Half-lines

A point A on a line divides the line excluding the point A into two disjointsubsets R and L such that

1. ifP, Q are two points in R or P, Q are points in L, then A is notbetween P and Q, and

2. ifP is in R and Q is in L, then A is between P and Q.

Figure 1.4: Half-lines

R can be regarded as the set of points whose coordinates are greaterthan the coordinate of A, and L the set of points whose coordinates areless than the coordinate of A.

Definition 1.6.1 Ray or Half-line: The two subsets R and L of theline, given above, are called rays or half-lines. A is called their originor initial point. Each of the two rays in a line having the same origin issaid to be the opposite of the other.

Let A and B be two points on a line. Then B belongs to one of thetwo rays into which the line is divided by A. That ray, containing B, is


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12 Points and Lines

called ray AB. The opposite ray may be called BA-produced. It mayalso be described as the set of points beyond A on lineAB.

By a geometric figure we mean a set of points. A pair of pointsdetermines six geometric figures:

1. lineAB

2. segAB, either open or closed

3. rayAB

4. rayBA

5. ABproduced6. BAproduced.

Figure (5) is the ray opposite to figure (4), and figure (6) is opposite tofigure (3).

Theorem 1.6.1 Given two points A, B and a ray with origin C, thereexists exactly one point D on the ray such that distCD = dist AB.

[Figure 1.5]

Figure 1.5: Line Segment and Ray

This is an inference from the Ruler Axiom 1.4.2 and the axiom 1.4.1of distance.


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1.7 Exercises 13

1.7 Exercises

1. Given that the coordinates of P and Q are 5 and 7 respectively,find

(a) distPQ

(b) dispPQ

(c) dispQP

(d) the coordinate of the midpoint ofP Q

2. P and Q are two points on the lines AB,CD respectively such thatdist AP = distCQ and distPB = dist QD. Can you assert thatdist AB = distCD ? Why ?


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14 Points and Lines


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Chapter 2

Convexity, Half-planes

and Angles

2.1 Convex Set

We intuitively know that the interior of a triangle or an angle is convex.1

In this section we give a formal definition of the concept of the interior ofgeometric figures, such as a segment, an angle and a triangle. We startwith the interior of a line segment and what is called half-planes.

Definition 2.1.1 Interior of a Segment: The open segment AB 2is

called the interior of seg AB. If seg AB itself is open, its interior isseg AB itself.

A figure which has no indent or notch in its boundary is said to beconvex in common parlance. Many geometric figures, like triangle, circleand square, are regarded as convex. The truth of it depends on the waywe define convexity. A formal definition of convexity is given below.

Definition 2.1.2 Convex Set: A set of points S is said to be convexif for any two points P and Q in S, segPQ lies in S. In symbols, S isa convex set if for any two points P, Q S, segPQ S.Some examples of convex sets are segment, ray, an angle and its interior,and a triangle and its interior. We are yet to define interior. For its

1See definitions 2.2.2 and 2.2.12See definition 1.5.2


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16 Convexity, Half-planes and Angles

definition we require an axiom.

Axiom 2.1.1 Axiom of half-planes: A line l in a plane divides theplane, excluding the line, into two disjoint subsets such that

1. each subset is a convex set, and

2. if P is in one of the subsets and Q in the other, then segPQintersects the line l.

Definition 2.1.3 Half-plane: The two subsets of the plane, describedin the axiom 2.1.1, are called half-planes. The line l is called the bound-ary of each of the half-planes. The half-planes are said to be on oppositesides of their common boundary.

The following theorem is an immediate inference from the above axiom.

Theorem 2.1.1 If three points A , B, C and a line l are in a plane suchthat the line does not pass through any of the three points, then eitherl intersects two of the three segments seg AB, seg BC, seg CA or it does

not intersect any of them.

[Figure 2.1] For, either all the three points A , B, Care in one of the half-

Figure 2.1: Points in Half-planes

planes bounded by l, or two of them are in one of the half-planes and thethird in the other.

Theorem 2.1.2 Intersection of two convex sets is convex.


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2.2 Geometric Figures 17

Figure 2.2: Intersection of Convex Sets

Proof [Figure 2.2] Let S1 and S2 be two convex sets and P and Qbe any two points in their intersection S1

S2. By definition 1.1.3 ofintersection,

P, Q S1Since S1 is convex, segPQ S1 (2.1)

Similarly,segPQ S2 (2.2)

By (2.1) and (2.2),

segPQ S1

S2 (2.3)

By definition 2.1.2, S1S2 is a convex set.

2.2 Geometric Figures

Just as a set may be regarded as the union of two or more sets,3 a figuremay be regarded as the union of two or more figures called sub-figures.Now, we give formal definitions of some of the familiar geometric figures.

Definition 2.2.1 Angle: A figure formed by the union of two rays, witha common origin, is called an angle. The common origin is called thevertex and the rays are called the sides of the angle. If the rays areopposite to each other, then the angle is called a straight angle. If therays are the same, the angle is called a null angle.

3See Defn. 1.1.5


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Notation. AOB denotes the angle whose vertex is O and sides areray OA, and ray OB. In symbols,

AOB = ray OA

ray OB

Definition 2.2.2 Triangle: A figure formed by the union of three non-collinear points and the segments joining them is called a triangle. Thethree points are called the vertices and the segments are called the sides ofthe triangle. If A , B, Care the vertices, CAB, ABC, andBCA arecalled the angles of the triangle which are briefly denoted byA,B,C,

respectively.

In symbols, ABC = seg AB seg BC seg CAMathematicians insist on a precise definition of the interior of a ge-

ometric figure like the angle or triangle, though a layman may not seeany need for that. We shall define the interior of an angle in terms ofhalf-planes.

Definition 2.2.3 Interior of an Angle: LetAOB be an angle whichis not straight nor null. The point B is in one of the half-planes boundedby line OA andA is in one of the half-planes bounded by lineOB. LetS1denote the half-plane in whichB lies andS2 denote the half-plane in whichA lies. The intersection S1

S2 is called the interior ofAOB.[Figure

2.3]

Figure 2.3: Interior of an Angle


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2.2 Geometric Figures 19

Since half-planes are convex sets, the interior of an angle is a convexset. Any angle is the boundary of its interior. IfP and Q are two pointsin the interior of an angle, then seg P Q, ray OP and ray OQ lie in theinterior.

Definition 2.2.4 Interior of a Triangle: The interior of a triangleis the intersection of the interiors of its angles.A triangle is the boundary of its interior.

Let A,B,C,D be four points in a plane, no three of which are collinear.The four points can be connected by six segments as in figure 2.4. In (b)

Figure 2.4: Segments Joining Four Points

of these figures no two of the six segments intersect, except at an endpoint. In (a) two segments AC,BD intersect. Out of the six segmentswe can always select four such that they form a closed chain of segmentswithout crossing one another.

Definition 2.2.5 Quadrilateral:

Let A,B,C,D be four points such that

1. no three of these are collinear, and

2. no two of the four segments AB,BC,CD,DA intersect, except atan end point.Then the union of the four segments is called quadrilateral ABCD.

In figure 2.5, figures (a) and (b) are quadrilaterals. Figure (c) is not aquadrilateral.


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Figure 2.5: Quadrilaterals

A Note on the Convexity of a QuadrilateralAccording to our definition 2.2.5 of a quadrilateral as the union of four

segments, no quadrilateral can be convex. But sometimes it is called con-vex if its interior is convex. In this sense, in Figure 2.5, the quadrilateral(a) is convex and (b) is not.

Problem 2.2.1 Five points in a plane can be connected by ten segmentsjoining them in pairs [Figure 2.6]. Can you choose five points in a plane

such that no two of the ten segments joining them intersect, except at anend Point ?

2.3 The Protractor Axiom

We have already defined an angle and its interior. Let AOB be an

angle which is not a straight angle nor a null angle. Let P be any Pointin the open segment AB. [Figure 2.7] Then ray OP is in the interior ofAOB. On the other hand, ifQ is any point on AB-produced or on BA-produced, then ray OQ is in the exterior of the angle. Also, if ray OC isin the interior ofAOB then it will intersect the open seg AB.

Angles are usually measured in degrees with a protractor. The basicaxioms regarding the measurement of angles are given below.


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2.3 The Protractor Axiom 21

Figure 2.6: Pentagon

Figure 2.7: Angles and Rays

Axiom 2.3.1 Protractor Axiom To every angle AOB, there corre-sponds a nonnegative number, called the measure of the angle, denoted bymAOB such that

1. mAOB > 0, ifAOB is not null,

2. mAOB = 0 iffAOB is null, and3. mAOB = mBOA.

Axiom 2.3.2 All straight angles have the same measure.

Notation: Measure of a straight angle is denoted by m st.


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Axiom 2.3.3 If ray OC is in the interior ofAOB, then mAOC +mCOB = mAOB.[Figure 2.8]

Figure 2.8: Sum of Angles

Axiom 2.3.4 IfAOA is a straight angle, andray OB is any ray, thenmAOB + mBOA = mAOA = m st, which is a constant by axiom2.3.2.[Figure 2.9]

Figure 2.9: Straight Angle

Axiom 2.3.5 GivenAOB and a line OA, there exists one and onlyone rayOB, in a given half-plane of which the line OA is the boundary,such that mAOB = AOB.[Figure 2.10]


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Figure 2.10: Equal Angles

We infer from the above axioms that the measure of an angle is nevergreater than the measure of a straight angle.4 We can choose the measureof a straight angle as a convenient number, which, of course, fixes theunit of angle. Two different measure numbers of a straight angle are incommon use. They are 180 and ( = 3.1416, correct to 4 decimals).The corresponding units of angle are called the degree, and the radian.Thus m st = 180 degrees = radians .

Notation. 180 degrees is denoted by 180o

and radians by c

. Radianmeasure is called circular measure also.

Definition 2.3.1 Supplementary Angles: Ifray OA andray OA areopposite rays and ray OB is any other ray, thenAOB andBOA aresaid to form a linear (supplementary) pair.[Figure 2.11]

If the measures of two angles are equal, the angles are said to be equal.Hereafter, it is in this sense that we make statements like A = B. Ifthe sum of two angles is equal to the measure of a straight angle, thenthe angles are said to be supplementary and each is called a supplementof the other.

Definition 2.3.2 Right angle: If the angles of a linear pair are equal,

4This assertion is true only in the present context. An extended definition of anglemeasure is given in the next section


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each is called a right angle. An angle less than a right angle is calledacute, and an angle greater than a right angle is called obtuse. If thesum of two angles is equal to a right angle, the angles are said to becomplementary.

Two angles are said to be adjacent if they have the same vertex and acommon side and if the remaining sides of the angles are in opposite half-planes bounded by the common side. In figure 2.8, AOC and BOCare adjacent.

Two angles are said to be vertically opposite, if the sides of one are

the rays opposite to the sides of the other. In figure 2.11, AOB andDOC are vertically opposite.

Theorem 2.3.1 Vertically opposite angles are equal in measure.

Figure 2.11: Vertically Opposite Angles

Given. In figure 2.11, AOB and DOC are vertically op-posite.Conclusion. mAOB = mDOC

Proof [Figure 2.11] AOB and BOD form a linear pair, and so

mAOB + mBOD = 180o (2.4)

Similarly,mDOC + mBOD = 180o (2.5)

The conclusion follows from (2.4) and (2.5).

Definition 2.3.3 Perpendicular Lines: Lines AB and P Q meetingat O are perpendicular ifAOP is a right angle.If O is the midpoint of seg AB also, then line P Q is called the perpen-dicular bisector of seg AB.


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Types of AnglesLet mAOB = o, then AOB is

1. an acute angle if 0 < < 90o

2. a right angle if = 90o

3. an obtuse angle if 90o < < 180o

4. a straight angle if = 180o

Theorem 2.3.2 Equality of Line Segments and Angles. Givenseg AB and seg CD, there exists just one point E on ray AB such that

AE = CD5

Also,

1. AB > C D if E is between A and B

2. AB = CD if E coincides with B

3. AB < C D if E is beyond B [Figure 2.12]

Figure 2.12: Equality of Line Segments

Theorem 2.3.3 Let A , B, C be three collinear points and O be a pointnot on the line containing them. Then ray OC is between the rays,ray OA and ray OB, iff C is between A and B.[Figure 2.13]

5Here AE stands for distAE. Hereafter we may stick to this practice when thereis no room for confusion


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Figure 2.13: Betweenness of Rays and Points

Example 2.3.1 In Figure 2.14, lines XY and M N intersect at O. IfP OY = 90o and a : b = 2 : 3, find c.

Figure 2.14: Example 2.3.1

Since a : b = 2 : 3, let

1. a = 2x, b = 3x.Since mP OY = 90o,

2. mP OX = 90o [Linear pair]


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2.4 Pencils and Quadrants 27

From steps (1) and (2),

a + b = 90o

5x = 90o x = 18o a = 36o, and

b = 54o

Since XOM and XON form a linear pair,

b + c = 180o

c = 180o 54o= 126o

2.4 Pencils and Quadrants

Consider several angles with the same vertex. Their sides form a set ofrays with the same origin.

Definition 2.4.1 Pencil at a Point: Let O be a point in a plane P.The set of all the rays inPwithO as origin is called the pencil determinedby O in P. The point O is called the vertex of the pencil. Any subset ofthe pencil at O is often referred to as a pencil at O.

Definition 2.4.2 Quadrants: Let OX,OY be two perpendicular raysof a pencil and OX, OY be their opposite rays. These four rays dividethe plane of the pencil, excluding the four rays, into four subsets knownas quadrants. They also divide the pencil at O, excluding the four rays,into four quarter pencils lying in the four quadrants.[Figure 2.15]

Let us call ray OX the initial ray of the pencil. Let ray OP be any

other ray of the pencil. Then ray OP may be regarded as the terminalposition of a ray which started rotating about O from the initial rayray OX. The rotation may be in one of two opposite directions. Thedirection of rotation from ray OX to ray OY through the interior of theright angle XOY is said to be positive or anti-clockwise. The oppositedirection is said to be negative or clockwise. Usually ray OX is taken ashorizontal and directed to the right and ray OY as vertical and directedupward.


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Figure 2.15: The Four Quadrants

2.5 Exercises

1. Given that AOB and BOC are adjacent angles, prove thatmAOB + mBOC = mAOC


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Chapter 3

Congruent Figures

3.1 Relations

Definition 3.1.1 Cartesian Product: If A, B are two sets, then theset of ordered pairs 1 (a, b) of elements a A and b B is called theCartesian Product of A an B, denoted by A B.

Example 3.1.1 If A = {a,b,c} and B = {c, d},then A B = {(a, c), (a, d), (b, c), (b, d), (c, c), (c, d)}

Definition 3.1.2 Relation: A subset R of A B is called a relationfrom A to B. If (a, b) R, we say a relates to b or b is related to a.Definition 3.1.3 Function: The relation R is a function from A to Bif every element in A relates to a unique element in B, and we usuallywrite f for R. If (a, b) f, we write b = f(a) and call b the imageof a. Function f is one-one(or injective) if distinct elements in A havedistinct images. Function f is onto(or surjective) if every element in Bis an image of some element in A.

Notation. A function f from A to B is written as

f : A B.

Example 3.1.2 If A = {1, 2, 3, 4} and B = {1, 4, 9, 16, 25},then R = {(1, 1), (2, 4), (3, 9), (4, 16)} is a relation from A to B.

1When we write {a, b}, the order in which the elements are written is insignificant.But in the ordered pair (a, b), a and b cannot be interchanged.


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30 Congruent Figures

In this example, if (x, y) R then you may note that y = x2 and hencethe relation can be written as R =

{(x, x2) : x

A

}. Since every element

in A relates to a single element in B, the relation is a function f, say.Thus f : A B is given by the formula f(x) = x2, x A. The functionf is one-one since no two distinct numbers in A have the same square. Itis not onto since 25 B is not the image(or square) of any element in A.

Definition 3.1.4 One-one Correspondence or Bijection: A func-tionf : A B is bijective if it is one-one and onto. If there is a bijective

function from A to B, we call it a one-one correspondence or bijectionbetween A and B.

Example 3.1.3 1. In example 5.1.1 if the element 25 is excluded fromthe set B, i.e. if B = {1, 4, 9, 16}, then the relation becomes a one-onecorrespondence between A and B.

2. There is a one-one correspondence between the set of students inyour class and the set of roll numbers allotted to them.

3. Birkhoff s Ruler Axiom establishes a one-one correspondence be-tween the setR of real numbers and the set of points on a line.

3.2 Introducing Congruence Relation

When we say that two figures are congruent, we mean that they have thesame shape and the same size, as the figures in 3.1.

What do we mean by the same shape and the same size? We assertthat

1. two segments are congruent if they have the same length, and

2. two angles are congruent if they have the same measure.

But the idea of congruence is not so simple when each figure consistsof many simple parts like segments and angles. A natural suggestion isto break up the figures into simple parts and to compare the parts of

one with the parts of the other. This brings in the idea of matchingup the parts of one with the parts of the other. The parts must agree innumber, in kind and in size. This matching up scheme is called a one-onecorrespondence. [Defn. 3.1.4]

In the light of the above discussion, we shall define congruence forfigures consisting of line segments and angles only.


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3.3 Congruence of Triangles 31

Figure 3.1: Congruent Figures

Definition 3.2.1 Congruent Figures: Two figures are congruent if

there is a one-one correspondence of the parts of the figures such thatcorresponding parts are congruent.

Now let us consider two triangles ABC and DEF. For congru-ence, there should be a one-one correspondence of the vertices. Such acorrespondence induces a correspondence of the angles and the sides ofthe triangles.

3.3 Congruence of Triangles

Definition 3.3.1 Congruent Triangles: Two triangles ABC andDEF are said to be congruent if there is a one-one correspondenceABC DEF such that AB = DE,BC = EF,CA = F D. andA =D,B = E,C = F.

The notation ABC = DEF means that the correspondence ABC DEF is a congruence. Hence the equivalence:2

ABC = DEF AB = DE,AC= DF,BC= EF,A = D,B =E,C = F

The six equations on the right of the above equivalence are not inde-pendent of each other. The theory of congruence of triangles is a study of

2For the meaning of equivalence, see definition 12.1.2


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the modes of dependence of those equations. Now we require an axiom.

Axiom 3.3.1 SAS Congruence AxiomA correspondence between twotriangles is given. If two sides and the included angle of one triangle areequal to the corresponding parts of the other, then the correspondence isa congruence.

Given. ABC DEF and AB = DE,AC = DF,A =DConclusion.

ABC

=

DEF 3

Theorem 3.3.1 ASA Congruence Theorem. A correspondence be-tween the vertices of two triangles is given. If two angles and the includedside of one triangle are equal to the corresponding parts of the other, thecorrespondence is a congruence.

Given. ABC DEF,A = D,B = E,AB = DE

Conclusion. ABC = DEF

Proof [Figure 3.2] Let P be a point on ray AC such that

AP = DF

Consider the correspondence ABP DEF.Since AB = DE,A = D,AP = DF, by SAS Axiom 3.3.1

ABP = DEF (3.1) ABP = E[Correspondingangles]

E = B[Given]

In ABC,ABP = ABC (3.2)

If AP < AC [Figure 3.2(a)], then ray BP divides internally ABC, andhence

ABP < ABC,

a contradiction of (3.1). IfAP > AC [Figure 3.2(c)], then rayBCdividesinternally ABP, and hence

ABP > ABC,

3SAS stands for side-angle-side.


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Figure 3.2: ASA Congruence

again a contradiction of (3.1). Therefore

AP = AC

P = C (3.3)

The conclusion follows from (3.1) and (3.3).The following theorem is stated without proof.

Theorem 3.3.2 AAS Congruence Theorem. A correspondence be-tween the vertices of two triangles is given. If two angles and any side ofone triangle are equal to the corresponding parts of the other, the corre-spondence is a congruence.

Theorem 3.3.3 Angles opposite to equal sides of an isosceles triangleare equal.

Given. ABC,AB = ACConclusion. B = C


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Figure 3.3: Angles Opposite to Equal Sides

Proof [Figure 3.3] Let the bisector ofA meet BC at D. ConsiderBAD CAD

AB = AC[Given]Since ray AD bisects A, BAD = CAD

Side AD is common. By the SAS theorem 3.3.1,

BAD = CAD ABD = ACD

Hence the conclusion.

Theorem 3.3.4 Converse of Theorem 3.3.3 The sides opposite toequal angles of a triangle are equal.

Given. ABC,B = C.Conclusion. AB = AC


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Proof [Figure 3.3] Let ADBC. Consider BAD CAD

B = C[Given]Since ADBC, BDA = CDA

Side AD is common. By the AAS Congruence Theorem 3.3.2,

BAD = CAD AB = AC

Example 3.3.1 In Figure 3.4, AD = BC,

DAB =

CBA. Prove that1. ABD = BAC2. BD = AC

3. ABD = BAC.


Proof Consider the correspondence ABD BACAD = BC

and DAB = CBA[Given]

AB = BA[Common side]


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and P M is the common side. Therefore, by SAS Congruence Axiom3.3.1,

AM P = BM P AP = BP[Corresponding sides]

The converse of this theorem is stated and proved in Chapter 11.[The-orem 11.2.1]

Example 3.3.2 ABC andDBC are isosceles, on the same base BCas in Figure 3.5. Segment AD is extended to intersect BC at P. Showthat

1. ABD = ACD2. ABP = ACP3. ray AP bisects A andD

4. line AP is the perpendicular bisector of seg BC.

Solution

1. Consider ABD ACDBy hypothesis, AB = AC,BD = CDAD common side.By SSS congruence theorem 3.3.5,

ABD = ACD (3.4)

2. Consider ABP ACPAB = AC [Given]AD common side.By (3.4), BAP = CAPBy SAS congruence 3.3.1,

ABP = ACP (3.5)

3. By (3.4) ray AP bisects A and D

4. By (3.5)

BP A = CP A

= 90o (3.6)

and BP = CP (3.7)

Therefore line AP is the perpendicular bisector of seg BC.


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Example 3.3.3 Let us consider two congruent figures consisting primar-ily of four points each. What is the meaning of fig ABCD = fig P QRS? It means that

1. A,B,C,D correspond to P,Q,R,S, respectively and

2. corresponding segments are equal in length, i.e. AB = PQ,AC =PR,AD = P S, etc.

As a consequence of the above two conditions, corresponding angles areequal, i.e. ABC = PQR,BDA = QSP etc.If fig ABCD = fig P QRS, we can infer that

1. ABC = P QR


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3.4 Exercises 39

2. seg AD = segPS

3. ACD = PRS,and similar results.

If two figures are congruent, corresponding sub-figures are congru-ent. In particular we choose segments and angles as sub-figures. Cor-respondence is a fundamental concept in mathematics. Congruence iscorrespondence with equality of corresponding parts.

If a congruence of two figures is given or established, any sub-congruence(as equality of segments, angles) can be written without any further ex-amination of the figures.

3.4 Exercises

Prove the following:

1. If segAB = segAB and X is a given point on line AB, thereis one and only one point X on line AB such that figABX =figABX

2. If ABC = ABC and X is a point on line AB, there is oneand only one point X on line AB such that

(a) figABX= figABX and

(b) figABCX= figA

B

C

X

3. If two triangles are congruent, their corresponding medians andattitudes are equal.


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Chapter 4

Triangles and Parallel

Lines

4.1 Exterior Angle Theorem.

In this section we study a basic theorem which is the source of all in-equalities in geometry. It is also a basis for the introduction of parallellines.

Note. Hereafter we may write AB for seg AB, dist AB, disp AB,and lineAB wherever the meaning is obvious from the context.

Definition 4.1.1 Exterior Angle: An angle forming a linear pair withan angle of a triangle is called an exterior angle.

In figure 4.1, ACD is an exterior angle ofABC.

Theorem 4.1.1 Exterior Angle Theorem. Each exterior angle of atriangle is greater than each of the remote (opposite) interior angles.

Restatement Consider ABC. IfBC is produced to D, then ACD >AProof[Figure 4.1] Let E be the midpoint of AC. Then

AE = EC

Let F be the point on BE-produced such that

BE = EF


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42 Triangles and Parallel Lines

Figure 4.1: Exterior Angle

Consider the correspondence AEB CEF.By theorem 2.3.1,

AEB = CEF

By the SAS axiom 3.3.1,

AEB = CEF (4.1) A = ECF (4.2)

Since rayCF lies in the interior ofACD,

ECF < ACD (4.3)

By (4.2) and (4.3),A < ACD

Note. In Figure 4.1, BF and CF are added to the figure representingthe data. This additional part of the figure is called construction. Con-struction is part of the proof and so there is no need to give a separateheading for construction, as it is sometimes done.


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4.1 Exterior Angle Theorem. 43

Corollary 4.1.2 An exterior angle of a triangle is equal to the sum ofinterior remote angles.

In figure 4.1, ACD = A +B

Corollary 4.1.3 If a triangle has a right angle or an obtuse angle, thenthe other two angles are acute.

Corollary 4.1.4 There cannot be more than one perpendicular to a linefrom an external point.

Theorem 4.1.5 If one side of a triangle is longer than another side, thenthe angle opposite to the longer side is bigger than the angle opposite tothe shorter side.

Theorem 4.1.6 Converse of Theorem 4.1.5 If one angle of a trian-gle is bigger than another angle, then the side opposite to the bigger angleis longer than the side opposite to the smaller angle.

The proof of this theorem depends on the exterior angle theorem 4.1.1.

Theorem 4.1.7 Triangle Inequality. The sum of the lengths of anytwo sides of a triangle is greater than the length of the third side.

Given. ABCConclusion. AB + AC > BC

Proof[Figure 4.2] Produce BA to D such that

AD = AC (4.4)

By the Converse of Theorem 3.3.3, in ACD,ADC = ACD (4.5)

Then, BCD > ACD

= BDC,by (4.5) (4.6)

By Theorem 4.1.5,

BD > BC

BA + AD > BC BA + AC > BC, by (4.4)


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Figure 4.2: Triangle Inequality

4.2 Parallel Lines

and Euclids Axiom

If we draw two lines on a sheet of paper, they usually meet at a point,provided the paper is big enough. This means that two coplanar linesare often intersecting. Are there non-intersecting coplanar lines? Yes,the lines in a ruled note book, you may say, are non-intersecting. Thisanswer is only suggestive.

Lemma 4.2.1 If two coplanar lines are perpendicular to a given line,then they are non-intersecting.

Given. Two coplanar lines l and m are perpendicular to agiven line P Q at two points A and B.


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4.2 Parallel Linesand Euclids Axiom 45

Conclusion. Lines l and m are non-intersecting.

Figure 4.3: Non-intersecting Lines

Proof[Figure 4.3] Suppose the lines l and m intersect at C.In ABC, exterior angle P AC and the remote interior angle CBAare right angles and therefore equal.This contradicts Theorem 4.1.1. Therefore our supposition is false. i. e. thetwo lines are non-intersecting.

Definition 4.2.1 Parallel Lines: Two lines are said to be parallel, ifthey are co-planar and do not intersect.

Theorem 4.2.2 Given a line and a point not on the line, there existsanother line through the given point and parallel to the given line.

Proof[Figure 4.4] Let A be the given point and m be the given line.Draw seg AB perpendicular to m. Draw line l perpendicular to AB at


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Figure 4.4: Parallel Line Through a Point

A, and lying in the plane of A and m. Then l and m are parallel, as

they are both perpendicular to AB. i. e. l is parallel to the given linem through the given point A.[By Lemma 4.2.1]

An axiom called Euclids Parallel Axiom or Euclids Fifth Postulateis stated below. Many important theorems in Euclidean geometry cannotbe proved without this axiom.

Axiom 4.2.1 Euclids Parallel Axiom.1 Through a given point thereis at most one line parallel to a given line.

Combining this axiom with the theorem proved above, we state the fol-lowing theorem.

Theorem 4.2.3 Given a line and a point not on the line, there is oneand only one line through the point and parallel to the line.

Definition 4.2.2 Transversal of Lines: Let a line EF cut two given

lines AB and CD at two distinct points, as in the figure 4.5. Line EFis called a transversal of lineAB and lineCD.

The eight angles in the figure, at the points of intersection, can beclassified as outer and inner angles with reference to the lines AB andCD.

1Euclid of Alexandria (300 BC), a Greek mathematician known as the Father ofGeometry.


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Figure 4.5: Transversal

The four angles lying between the lines AB and CD shall be calledinner angles, and the remaining four shall be called outer angles. If wecount the inner angles in a cyclic order 1 2 3 4 1,then 1,3 are alternate in the counting, so also are 2,4.

The angles counted alternately are called alternate angles. A pairof alternate angles are non-adjacent inner angles on opposite sides of thetransversal. Each inner angle corresponds to an outer angle. A pairof corresponding angles are non-adjacent angles on the same side of the

transversal, one being inner and the other outer.Thus in the figure 1,3;2,4 are pairs of alternate angles: 1,1;2,2;3

are pairs of corresponding angles.

Axiom 4.2.2 Corresponding Angles AxiomIf a transversal cuts twoparallel lines, then any two corresponding angles are equal.

Theorem 4.2.4 Converse of Axiom 4.2.2 If a transversal cuts two


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coplanar lines such that a pair of corresponding angles are equal, then the

two lines are parallel.

Though in some books this converse is stated as an axiom, here we stateand prove it as a theorem.

Figure 4.6: Corresponding Angles

Given. Transversal P S cuts coplanar lines AB and CD atQ and R respectively. P QB = QRDConclusion. Lines AB and CD are parallel.

Proof[Figure 4.6] Suppose lines AB and CD are not parallel.Let the lines intersect at a point, say T.By the Exterior Angle Theorem 4.1.1 applied to RT Q,

P QB = QRDThis contradicts the hypothesis. Therefore, lines AB and CD are parallel.


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Theorem 4.2.5 If a transversal cuts two parallel lines, then any two

alternate angles are equal.

Given. Lines AB and CD are parallel. Line P S cuts ABand CD at Q and R, respectively.

Conclusion. BQR = CRQ,AQR = DRQ

Figure 4.7: Alternate Angles

Proof[Figure 4.7] By Axiom 4.2.2,

CRQ = AQP (4.7)

By Theorem 2.3.1,BQR = AQP (4.8)

From (4.7) and (4.8),BQR = CRQ

Similarly,AQR = DRQ

Theorem 4.2.6 Converse of Theorem 4.2.5 If a transversal cuts


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two coplanar lines such that two alternate angles are equal, then the two

lines are parallel.

Given. Line P S cuts AB and CD at Q and R respectively.BQR = CRQConclusion. Lines AB and CD are parallel.

Proof[Figure 4.7] By hypothesis,

BQR = CRQBy Theorem 2.3.1,BQR = AQP

Therefore,CRQ = AQP

The conclusion follows from the Converse of Axiom 4.2.2.

Note. Another proof, similar to that of Theorem 4.2.4, can also begiven to the above theorem.

Theorem 4.2.7 The sum of the angles of a triangle is equal to two rightangles.

Given. ABCConclusion. A +B + C = 180o

Proof[Figure 4.8] Let line P Q BC, pass through A as in Figure 4.8.Since the transversal lineAB cuts the parallel lines BC and P Q,

B = P AB [Alternate angles]

Similarly, C = QAC

Since PQ is a line,

P AB + BAC+QAC = 180o

The conclusion follows from the above three steps.

No two sides of a triangle can be parallel. But a quadrilateral canhave parallel sides. Different types of quadrilaterals are listed below.

1. A quadrilateral is a parallelogram if its opposite sides are parallel.

2. A parallelogram is a rhombus if all its sides are equal.

3. A parallelogram is a rectangle if all its angles are right angles.

4. A rectangle is a square if all its sides are equal.

5. A quadrilateral is a trapezium if it has a pair of parallel sides.


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Figure 4.8: Sum of Angles of a Triangle

Theorem 4.2.8 A diagonal of a parallelogram divides it into two con-gruent triangles.

Given.Parallelogram ABCD. Diagonal AC.Conclusion. ABC = CDA

Proof[Figure 4.9] Consider ABC CDA.Since BC AD and lineAC is a transversal,

BCA = DAC [Alternate angles] (4.9)

Since AB DC and lineAC is a transversal,

BAC = DCA [Alternate angles] (4.10)

AC = CA, Common side(4.11)


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Figure 4.9: Diagonal of a Parallelogram

The conclusion follows from (2), (3), (4) and ASA Congruence Theorem

3.3.1. The following corollaries can be proved as in the above theorem.

Corollary 4.2.9 Opposite sides of a parallelogram are equal.

Corollary 4.2.10 If opposite sides of a quadrilateral are equal, then itis a parallelogram.

Corollary 4.2.11 Opposite angles of a parallelogram are equal.

Corollary 4.2.12 If opposite angles of a quadrilateral are equal, then itis a parallelogram.

Theorem 4.2.13 If two opposite sides of a quadrilateral are equal andparallel, then it is a parallelogram.

Theorem 4.2.14 The diagonals of a parallelogram bisect each other.


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Figure 4.10: Diagonals of a Parallelogram

Given.Parallelogram ABCD. The diagonals AC and BDmeet at O.Conclusion.AO = CO and BO = DO

Proof[Figure 4.10] Consider the correspondence AOB COD SinceAB DC and line AC is a transversal,

BAO = DCO [Alternate angles] (4.12)

Similarly,

ABO = CDO (4.13)

AB = DC Opposite sides of a parallelogram (4.14)

By ASA Congruence Theorem 3.3.1,

AOB = COD (4.15)AO = CO (4.16)

BO = DO Corresponding sides (4.17)


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Example 4.2.1 In Figure ??, P Q P S , P Q SR,SQR = 28o, andQRT65o. Find x and y.

1. In PQS,SP Q = 90o [Given]2. Since the sum of angles of a triangle is 180o, x + y = 90o

3. Since P Q

SR, and line SQ is a transversal, RSQ = x

4. Since QRT is an exterior angle of SRQ, T RQ = RSQ +28o 65o = x + 28o x = 37o.

5. Substituting in (2), y = 53o

4.3 Polygons

Let Ai, i = 1, 2, n, n > 2 be n points in a plane, no three of whichare collinear. The union of the n line segments A1A2, A2A3, AnA1 iscalled a polygon with n sides (or n-gon) if no two of them intersect exceptat the end points. A triangle is a 3 gon, a quadrilateral a 4 gon, anda pentagon a 5 gon.

In a polygon, the points are called vertices and the line segments arecalled the sides. Two vertices are adjacent (or consecutive) if they are

joined by a side. Two sides are adjacent(or consecutive) if they have acommon endpoint.

The segment joining two non-consecutive vertices is called a diagonal.A triangle has no diagonal. A quadrilateral has two, and a pentagon has5 diagonals.

A polygon is regular if all its sides are equal and all its angles areequal.

4.4 Exercises

1. If a transversal cuts two parallel lines, and if a pair of alternateangles are equal, show that

(a) the second pair of alternate angles also are equal, and

(b) any pair of corresponding angles are equal.


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4.4 Exercises 55

2. If a transversal cuts two parallel lines, and a pair of corresponding

angles are equal, show that

(a) any pair of corresponding angles also are equal, and

(b) any pair of alternate angles are equal.


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Chapter 5

Symmetric Figures and

Circles

5.1 Introduction

We know that mammals, birds, and fishes have bilateral symmetry. Thisseems to be a requirement for balance in rapid motion. Bilateral symme-try is maintained in the construction of cars, planes and ships. Some ofthe slow moving animals of the lower order, such as starfish and jellyfishand many flowers and plants have radial symmetry.

In plane geometry the typical examples of figures having bilateral andradial symmetries are the isosceles triangle and the regular polygons. Thiswill be clear when we formulate the concept of symmetry. Let us studya few examples.

Example 5.1.1 Isosceles triangle. A triangle is isosceles if two of itssides are equal. If ABC is isosceles with AB = AC, we can see that

ABC =

ACB

[Figure 5.1]

1. Consider the correspondenceABC ACB A A, B C.

2. Since AB = AC,AC = AB, and BC = CB, this correspondenceis a congruence, by the SSS theorem 3.3.5.

ABC

=

ACB


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58 Symmetric Figures and Circles

Figure 5.1: Isosceles Triangle

In the congruence ABC = ACB, A remains unaltered, B is car-ried to C and C is carried to B so that the midpoint of BC, say Mremains unaltered. A rotation of the triangle about the line AM through180o brings it back to its original position. For this reason line AM iscalled an axis of symmetry of the triangle. This type of symmetry iscalled axial symmetry.

Also ABM is the mirror image of ACM in the line AM. In thisexample we have taken AB = AC and hence the median(or altitude)

through A is the axis of symmetry.Let D , E , F be the midpoints of the sides BC,CA,AB of ABC.

Then, ABC is isosceles iff one of the following is true:

1. ABC = ACB [Line AD is the axis of symmetry.]2. BC A = BAC [Line BE is the axis of symmetry.]3.

CAB

=

CBA [Line CF is the axis of symmetry.]


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5.1 Introduction 59

This example shows that the congruence of a figure to itself and the

property of symmetry are closely related. This leads to the definition

Definition 5.1.1 Symmetry of Figures: Any congruence of a figureto itself is called a symmetry of the figure.

Every symmetry of a figure is a one-one correspondence of the figure toitself, which is also a congruence. Any figure is congruent to itself underthe correspondence which carries each point to itself. Such a congruence is

called the identity congruence or identity symmetry. This is also includedamong the symmetries by our definition. But we shall say that a figure issymmetric only if it has a symmetry different from the identity symmetry.

Example 5.1.2 Equilateral Triangle. A triangle is equilateral if allits sides are equal. Each of the altitudes of an equilateral triangle is anaxis of symmetry.

Figure 5.2: Equilateral Triangle


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[Figure 5.2] The triangle ABC is equilateral. It coincides with its original

position (let us say, coincides with itself), when it is rotated about itscentre through 120o or 240o. This means that the rotation carries A toB, B to C and C to A, or A to C, C to B and B to A. The figure hasthree axial symmetries and two cyclic symmetries. Each symmetry canbe expressed as a congruence. For example, ABC = ACB meansthat the triangle is symmetric about the median (or altitude) from A. Itis an axial symmetry. And ABC = BCA means that the trianglehas cyclic symmetry about its centre O.

The non-identity symmetries of the equilateral triangle in figure 5.2are listed below.

1. Cyclic Symmetries

(a) ABC = BCA [Rotation through 120o about the centre](b) ABC = CAB [Rotation through 240o about the centre]

2. Axial Symmetries

(a) ABC = ACB [Symmetric about line AD](b) ABC = CBA [Symmetric about line BE](c) ABC = BAC [Symmetric about line CF]

Example 5.1.3 Square A quadrilateral is a square if its sides are equaland its angles are equal.

[Figure 5.3] A square coincides with itself when it is rotated about itscentre through 90o, 180o, or 270o. It also coincides with itself when itis rotated thorough 180o about each of its diagonals or each of the linesbisecting the pairs of opposite sides.

In figure 5.3, l is the line bisecting the opposite sides AB,CD and mis the line bisecting the opposite sides AD,BC.

The square has 7 non-identity symmetries, each defined by a congru-ence. They are listed below.

1. Cyclic Symmetries

(a) ABCD = BCDA [Rotation through 90o about the centre O.](b) ABCD = CDAB [Rotation through 180o about the centre

O.]

(c) ABCD = DABC [Rotation through 270o about the centreO.]


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5.1 Introduction 61

Figure 5.3: Square

2. Axial Symmetries

(a) ABCD = ADCB] [About line AC](b) ABCD = CBAD [About line BD](c) ABCD = BADC [About line l](d) ABCD = DCBA [About line m]

Including the identity, an isosceles triangle has two, a square haseight, an equilateral triangle has six, and a rectangle has four symmetries.

Symmetries of more complicated figures can be studied on the basisof the above definition. The physicists and the chemists are interested inthe symmetries of the atoms and the molecules.1

1Note for Teachers: Groups of symmetries are important in Group Theory. Thegroup of symmetries of an equilateral triangle is isomorphic to the permutation groupS3 and that of a square is isomorphic to the Octic Group D4.


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5.2 Types of Symmetry

Definition 5.2.1 Axial Symmetry: A plane figure has axial symmetryabout a line L, if it coincides with itself when rotated through 180o aboutL.

Example 5.2.1 Have a look at Figure 5.4. The figure can be regarded asbeing divided by L into two parts, so that each part is the mirror imageof the other in the line L. If P is a point of the figure, there is another

point P

of the figure such that segPP

is bisected at right angles by L.

Figure 5.4: Axial Symmetry

The figure fg504 is the graph [Defn. ??] of the function f(x) = x2. They-axis is the line of symmetry. This symmetry makes the function even.

Definition 5.2.2 Central Symmetry: A plane figure has central sym-metry about a point O, if it coincides with itself when rotated through180o about O.


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5.2 Types of Symmetry 63

Example 5.2.2 Have a look at Figure 5.5. If P is a point of the figure,

there is another point P of the figure such that segPP is bisected at O.

Figure 5.5: Central Symmetry

The figure 5.5 is the graph off(x) = x3

. It is symmetric about the origin.This symmetry makes the function odd.[Defn. ??]

Definition 5.2.3 Cyclic Symmetry: A plane figure has cyclic sym-metry about a point O, if it coincides with itself when rotated through360

n

oabout O, where n is a positive integer. n is called the period of the

symmetry.

A regular polygon with n sides has cyclic symmetry with period n.


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Among the above examples, equilateral triangle and square are regular

polygons.

If the period of a cyclic symmetry is 2, it is a central symmetry.Symmetries occur in geometry and algebra, in nature and in the world

of ideas. The student should make the maximum use of symmetries.

5.3 Circles

A circle is the most symmetric of all the plane figures. It has axialsymmetry about any line passing through its centre. Rotation about thecentre through any angle makes a circle coincide with itself. So, it hasinfinite number of cyclic symmetries also. A precise definition of a circleis necessary before we study more of its properties.

Definition 5.3.1 Circle: A circle is the set of all points in a given planeeach of which is at a given distance from a given point of the plane. The

given point is called the centre and the given distance the radius of thecircle.

Let O be the centre and r the radius of the circle. The points of the planefall into three disjoint sets:

1. The set of points of the circle.

2. The set of points at a distance less than r from O

3. The set of points at a distance greater than r from O.

The first set is the circle, the second is called the interior or inside of thecircle, and the third its exterior.

Definition 5.3.2 Chord of a Circle: A chord of a circle is a segmentwhose end points are points of the circle. The line containing a chord iscalled a secant.

On any line through the centre O, there are just two points at a dis-tance r from O. The segment determined by the two points is a diameter.A diameter is a chord through the centre. A radius is a segment whoseend points are the centre and a point of the circle. All the radii are ofequal length, and that length is the radius that is defined in 5.3.1.

Let O be the centre of a circle of radius r. Let F be the foot of theperpendicular from O to a line l in the plane of the circle. [Figure 5.6]There are three possibilities:


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5.3 Circles 65

Figure 5.6: Tangent & Secant

1. OF > r. Then every point of the line is outside the circle.(Prove)

2. OF = r. Then the circle and the line have just one point in common.Every other point of the line is outside the circle (Prove). The lineis then said to be tangent to the circle at F. It is perpendicular tothe radius OF.

3. OF < r. Then the line and the circle intersect at two points, say Aand B. (Does this not require a proof ?) Line AB is a secant and

seg AB is a chord.

Theorem 5.3.1 The tangent at any point on a circle is perpendicular tothe radius through the point of contact.

Proof [The student is advised to draw the figure.] Let O be the centreof the circle, P be the point of contact of the tangent l. Then OP = r,the radius of the circle. Suppose Q is any point on l other than P. In


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the light of the above discussion,

OQ > r (5.1)

This means that OP is the shortest distance from O to the tangent l.So,

OP l (5.2)Theorem 5.3.2 The lengths of tangents from an external point to a cir-cle are equal.

Figure 5.7: Tangents from an External Point

Proof [Figure 5.7] Let O be the centre of the circle, P the external

point, and Q, R the points of contact of tangents from P.[The student isadvised to draw the figure.] Consider the correspondence P QO P ROBy Theorem 5.3.1, P QO and P RO are right angles. The radii,

OQ = OR (5.3)

segPO is a common hypotenuse. By RHS Congruence Theorem 3.3.6,

P QO = P RO (5.4)P Q = P R Corresponding sides (5.5)


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5.3 Circles 67


In a circle, the triangle formed by a chord and the radii joining itsend points, is isosceles. Many properties of a circle can be derived fromthis simple fact. This is the source of the symmetry of the circle.

Definition 5.3.3 Arcs and Angles: An angle at the centre of circle,called a central angle, cuts the circle at two points. Consider a centralangle AOB cutting the circle at A, and B.

[Figure 5.8]

Figure 5.8: Arcs and Angles

Part of the circle is in the interior ofAOB and part is in the exterior.The part in the interior together with the points A, B is called the minorarc AB or simply arcAB. The part in the exterior ofAOB togetherwith A, B is called the major arc AB. AOB is said to be the anglesubtended at the centre by arcAB. A and B are the end points ofarcAB.


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Definition 5.3.4 Semicircle: If seg AB is a diameter of a circle with

centre O, there are two arcs with end points A and B neither of whichcan be called major or minor. Each is called a semicircle.

Definition 5.3.5 Cyclic Quadrilateral: LetA,B,C,D be four pointson a circle such that no two of the four segments seg AB, seg BC, seg CD,seg DAinterest except at the end points. Then the figure ABCD is said to be acyclic quadrilateral.

Figure 5.9: Cyclic Quadrilateral

BAD is said to be inscribed in the arcBAD, and subtended byarcBCD at the circumference.[Figure 5.9]

Theorem 5.3.3 The angle subtended by an arc at the circumference isequal to half the angle subtended at the centre.

Given. arcPQ subtends P AQ at the circumference of acircle with centre O.


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5.3 Circles 69

Conclusion. P AQ = 12P OQ

Figure 5.10: Angles Subtended by an Arc

Proof [Figure 5.10] Extend seg AO to a point B. Since BOQ is anexterior angle ofAOQ, and OA = OQ, by Corollary 4.1.2,

BOQ = OAQ + OQA (5.6)

= 2 OAQ (5.7)Similarly,

BOP = 2 OAP (5.8)Adding (2) and (3) we get the conclusion.

Corollary 5.3.4 Angles inscribed in an arc are equal.

This is a direct consequence of the previous theorem.


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Corollary 5.3.5 Any angle inscribed in a semicircle is a right angle.

Corollary 5.3.6 IfBAC is a right angle, then A is on the circle withBC as diameter.

A few more theorems are stated without proof.

Theorem 5.3.7 Opposite angles of a cyclic quadrilateral are supplemen-tary.

Theorem 5.3.8 An angle of a cyclic quadrilateral is equal to the oppositeexterior angle.

Theorem 5.3.9 The angle between a tangent to a circle and a chordthrough the point of contact, is equal to the angle subtended by the chordat the centre of the circle.

Theorem 5.3.10 If a quadrilateral ABCD is circumscribed to a circle,then AB + CD = BC+ DA.

5.4 Exercises

1. Study the symmetries of

(a) a rectangle(b) a regular hexagon

2. A figure consists of five co-planar points A , B, C, D, O and the seg-ments joining them. The figure has a symmetry given by the con-gruence fig ABCDO = fig BCDAO. Show that fig ABCD is asquare and O is its centre.

3. A,B,C,D are four coplanar points. The figure formed by join-

ing the points in pairs, has a symmetry given by the congruencefig ABCD = fig BADC. Show that the figure is an isoscelestrapezoid or a parallelogram.

4. If a figure has cyclic symmetry of period 4, 6, 8, , show that ithas central symmetry. Consider the square, the regular hexagon,regular octagon etc.

5. Show that a regular pentagon has no central symmetry.


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5.4 Exercises 71

6. Show that all regular polygons have axial symmetry.

7. Show that a non-rectangular parallelogram has central symmetrybut no axial symmetry.

8. If a figure has two perpendicular axes of symmetry, show that ithas central symmetry.

9. Draw the pair of tangents to a given circle from a given point usingruler and compasses.

10. Draw the common tangents to two given circles.


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87/184

Chapter 6

Euclids Ratio Theorem.

6.1 Harmonic Division

We assume that you have already learnt some properties of ratios inalgebra and arithmetic. Here we are concerned with the applications ofratios to geometry.

Let P,A,B be distinct collinear points. The position of P on the lineAB can be described in several ways. Some of them are:

1. In terms of the coordinate of P referred to a coordinate system for

the line AB. [Definition 1.4.1].

2. In terms of distances of P from A and B and the distance betweenA and B.

3. In terms of a ratio referred to A and B.

Note: In this section, we denote disp AB byAB and dist AB by AB.

If the distances AP,PB, and AB are given, the ratio APPB

and the

betweenness relations of P with respect to A and B (i. e. whether P isbetween A and B or not) can be determined. Conversely, if the ratio AP

PB

and the betweenness relations of P relative to A and B are given, theposition of P can be determined.

Definition 6.1.1 Position Ratio: If A, B and P are three collinear

points, the ratio dispAPdisp PB

=APPB

is called the position ratio of P relative to

A and B. The numerical value of the ratio is APPB


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74 Euclids Ratio Theorem.

Figure 6.1: Position Ratio

The position ratio is positive if P is between A and B, and it isnegative if P is not between A and B.

Let us introduce a coordinate system in the line. It does not matterwhat the origin is. Let a,b,x be the coordinates of A,B,P, respectively.

Then AP = x a and P B = b x by definition...Therefore, position ratio of P = xa

bx

There is some advantage if we choose A as the origin. Then

AP

P B=

x 0b x (6.1)

=x

b x (6.2)

= , say, (6.3)

where is the value of the position ratio.If = 1, then we get b = 0, which is not possible since points A and Bare distinct. Therefore = 1.

Let us determine x in terms of .

x

b a = (6.4)

x = (b x) (6.5)(1 + )x = b (6.6)

x =b

1 + (6.7)

i. e. AP = x = b1+

, 1 + = 0Thus we have expressed the position ratio of P in terms of the co-

ordinate x of P and conversely the coordinate in terms of the position


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6.1 Harmonic Division 75

ratio.1

Definition 6.1.2 Internal and External Division: Let A, B and Pbe distinct collinear points, and dispAP

disp PB= .

1. When the ratio is positive, seg AB is said to be divided internallyin the ratio ||.

2. When the ratio is negative, seg AB is said to be divided externallyin the ratio

||.

Combining the two cases, we may say that seg AB is divided in theratio . It becomes external division when is negative. The funda-mental fact is the relation x

bx= , which connects position ratio and

coordinates.

Theorem 6.1.1 Given a segmentAB and a ratio , there exists one andonly one point P dividing seg AB in the ratio , provided

=

1.

Proof Let us suppose that there is a point P dividing seg AB in theratio . Let b, x be the coordinates of B, P with A as the origin. Then,

=

APP B

(6.8)

=x

b

x

(6.9)

=x

b x (6.10)

Equation (3) has one and only and solution

x =b

1 + , (6.11)

provided 1 + = 0. Therefore there is one and only one point P dividingseg AB in the given ratio and its coordinate is b

1+referred to the origin

A. Note

1. The solution x = b1+

is meaningless when 1 + = 0 i.e. when = 1. There is no point on the Euclidean line AB dividingseg AB in the ratio 1.

1The theory of ratio, in geometry, is essentially algebraic.


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2. The ratioAP

PB

= xbx

is not defined when x = b i. e. when P is at

B.

3. We have proved the unique existence of a point P dividing seg ABin the given ratio .

Theorem 6.1.2 If P divides seg AB in the ratio lm

, then,AP = l

ABl+m

,P B = m

AB

l+m

We present two proofs, the first without using and the second using thecoordinates.Proof(1) Let O be the origin of a coordinate system for the line AB.

SinceAB = disp AB, we have by Definition 1.4.2

AB =

OB OA (6.12)

By Theorem 1.4.1,AB =

AP +

P B (6.13)

By hypothesis,

APP B

=l

m(6.14)

Since ab

= cd

aa+b

= cc+d

,

AP

AP +

P B

=l

l + m(6.15)

APAB

=l

l + m(6.16)

AP = l ABl + m

(6.17)

Similarly,

P B =

mAB

l + m(6.18)

Proof(2) Choose A as the origin, and let b and p be the coordinates of


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B and P respectively.

AB = b (6.19)AP = p (6.20)P B = b p (6.21)p

b p =l

m(6.22)

mp = l(b

p) (6.23)

(l + m)p = lb (6.24)

p =lb

l + m(6.25)

AP =

lAB

l + m(6.26)

The rest of the proof is evident.

In this context l, and m may be positive or negative, provided onlyl + m = 0 or l

m= 1.

The student should also have a physical picture of the above solution.

WhenAPPB

= lm

, l > 0, m > 0 we may regard seg AB as being divided into

l + m equal parts of which segAP contains l parts and segPB containsm parts. In this physical picture we have restricted l, m to be positiveintegers.

Definition 6.1.3 Harmonic Division: If a segment AB is divided atC and D internally and externally (or externally and internally) in thesame ratio, then seg AB is said to be divided harmonically at C and D,or (AB,CD) is called a harmonic range. In this case we may simply saythat (AB,CD) is harmonic.

In other words, (AB,CD) is a harmonic range ifAC

CB

=

AD

DB

[Figure 6.2]

Theorem 6.1.3 If (AB,CD) is harmonic, then

1. (BA,CD) is harmonic, and

2. (CD,AB) is harmonic.

Proof


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Figure 6.2: Harmonic Division

1.

ACCB

= ADDB

Given (6.27)

CBAC

= DBAD

(6.28)

BCCA

= BDDA

, [P Q = QP] (6.29)

(BA,CD) is a harmonic range.

2.

ACCB

= ADDB

(6.30)

ACAD

= CBDB

(6.31)

CAAD

= CBBD

(6.32)

(CD,AB) is a harmonic range.

Note. When (AB,CD) is harmonic, the same relation is sometimesstated as (ACBD) is harmonic. In the first notation, there is a commaseparating the four letters into two pairs. In the second notation, thepairs are interlocked without a comma.

Theorem 6.1.4 If (AB,CD) is harmonic, then,2AB

= 1AC

+ 1AD

, and the converse.


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Note. This means thatAB is the harmonic mean between

AC and

AD, and hence the name harmonic range. Proof Choose A as the origin.Let the coordinates of B, C, D be b,c,d, respectively. Since (AB,CD) isharmonic

ACCB

= ADDB

(6.33)

c

b

c

= db

d

(6.34)

AC = disp AC (6.35)= c (6.36)

c(b d) + d(b c) = 0 (6.37)bc + bd 2cd = 0 (6.38)

Dividing each term in (3) by bcd,

2b

= 1c

+ 1d

(6.39)

2AB

=1

AC

+1

AD

(6.40)

The converse of the theorem holds, since the steps are reversible.

Two direct proofs of the converse is given below, first without usingand then using coordinates.

Theorem 6.1.5 Converse of theorem ?? If A,B,C,D are collinearpoints such that 2

AB= 1

AC+ 1

AD, then (AB,CD) is harmonic.

Proof(1) By hypothesis,

2AB

=1

AC

+1

AD

(6.41)

Multiplying each term of (1) by AB,2 =

ABAC

+

ABAD

=

AC+

CB

AC

+

AD +

DB

AD

= 1 +

CB

AC+ 1 +

DB

AD


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CB

AC +DB

AD = 0 (6.42)Therefore,

ACCB

= ADDB

(6.43)

By (3) and the definition 6.1.3 of harmonic range, (AB,CD) is harmonic.Proof(2) Choose A as the origin. Let b,c,d be coordinates of B, C, Drespectively, i. e.

AB = b (6.44)AC = c (6.45)AD = d (6.46)

By hypothesis, 2b

= 1c

+ 1d

Multiplying each term by b,

logical foundations of geometry

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