M. N. AREF AND WILLIAM WERNICK PROBLEMS & INSSOLUTIONS EUCLIDEAN GEOMETRY DOVER BOOKS ON MATHEMATICS HANDBOOK OF MATHEMATICAL FUNCTIONS: WITH FORMULAS, GRAPHS, AND MATHEMATICAL TABLES, Edited by Milton Abramowitz and Irene A. Stegun. (0-486-61272-4) ABSTRACT AND CONCRETE CATEGORIES: THE JOY OF CATS, Jiri Adamek, Horst Herrlich, George E. Strecker. (0-486-46934-4) NONSTANDARD METHODS IN STOCHASTIC ANALYSIS AND MATHEMATICAL PHYSICS, Sergio Albeverio, Jens Erik Fenstad, Raphael Hoegh-Krohn and Tom Lindstrom. (0-486-46899-2) MATHEMATICS: ITS CONTENT, METHODS AND MEANING, A. D. Aleksandrov, A. N. Kolmogo- rov, and M. A. Lavrent'ev. (0-486-40916-3) COLLEGE GEOMETRY: AN INTRODUCTION TO THE MODERN GEOMETRY OF THE TRIANGLE AND THE CIRCLE,NathanAltshiller-Court. (0-486-45805-9) THE WORKS OF ARCHIMEDES, Archimedes. Translated by Sir Thomas Heath. (0-486-42084-1) REAL VARIABLES WITH BASIC METRIC SPACE TOPOLOGY, Robert B. Ash. (0-486-47220-5) INTRODUCTION TO DIFFERENTIABLE MANIFOLDS, Louis Auslander and Robert E. MacKenzie. (0-486-47172-1) PROBLEM SOLVING THROUGH RECREATIONAL MATHEMATICS, Bonnie Averbach and Orin Chein. (0-486-40917-1) THEORY OF LINEAR OPERATIONS, Stefan Banach. Translated by F. Jellett. (0-486-46983-2) VECTOR CALCULUS,Peter Baxandall and HansLiebeck. (0-486-46620-5) INTRODUCTION TO VECTORS AND TENSORS: SECOND EDITION-TWO VOLUMES BOUND AS ONE, Ray M. Bowen and C: C. Wang. (0-486-46914-X) ADVANCED TRIGONOMETRY, C. V. Durell and A. Robson. (0-486-43229-7) FOURIER ANALYSIS IN SEVERAL COMPLEX VARIABLES, Leon Ehrenpreis. (0-486-44975-0) THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 1, Euclid. Edited by Thomas L. Heath. (0-486-60088-2) THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 2, Euclid. (0-486-60089-0) THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 3, Euclid. Edited by Thomas L. Heath. (0-486-60090-4) AN INTRODUCTION TO DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS, Stanley J. Farlow. (0-486-44595-X) PARTIAL DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS, Stanley J. Farlow. (0-486-67620-X) STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATIONS, Avner Friedman. (0-486-45359-6) ADVANCED CALCULUS, Avner Friedman. (0-486-45795-8) POINT SET TOPOLOGY, Steven A. Gaal. (0-486-47222-1) DISCOVERING MATHEMATICS: THE ART OF INVESTIGATION, A. Gardiner. (0-486-45299-9) LATTICE THEORY: FIRST CONCEPTS AND DISTRIBUTIVE LATTICES, George Gratzer. (0-486-47173-X) (continued on back flap) PROBLEMSANDSOLUTIONS IN EUCLIDEAN GEOMETRY PROBLEMSANDSOLUTIONS IN EUCLIDEAN GEOMETRY M. N. AREF Alexandria University WILLIAM WERNICK City College, NewYork DOVER PUBLICATIONS, INC MINEOLA, NEW YORK Copyright Copyright © 1968 by Dover Publications, Inc. All rights reserved. Bibliographical Note This Dover edition, first published in 2010, is a reissue of a work originally published by Dover Publications, Inc, in 1968. International Standard Book Number ISBN-13: 978-0-486-47720-6 ISBN-10: 0-486-47720-7 Manufactured in the United States by Courier Corporation 47720701 www.doverpublications.com Dedicated to my father who has been always of lifelong inspiration and encouragement to me M.N.A. CONTENTS PREFACE xi SYMBOLS EMPLOYED IN THIS BOOK xiii CHAPTER 1 - Triangles and Polygons Theorems and corollaries I Solved problems 4 Miscellaneous exercises 25 CHAPTER 2 - Areas, Squares, and Rectangles Theorems and corollaries 33 Solved problems 35 Miscellaneous exercises 56 CHAPTER 3 - Circles and Tangency Theorems and corollaries 64 Solved problems 67 Miscellaneous exercises 95 CHAPTER 4 -Ratio and Proportion Theorems and corollaries 105 Solved problems io8 Miscellaneous exercises 133 CHAPTER 5 Loci and Transversals Definitions and theorems 144 Solved problems 145 Miscellaneous exercises 169 CHAPTER 6 -Geometry of Lines and Rays HARMONIC RANGES AND PENCILS Definitions and propositions 178 Solved problems 182 ISOGONAL AND SYMMEDIAN LINES-BROCARD POINTS Definitions and propositions 186 Solved problems 189 Miscellaneous exercises 191 ix X CONTENTS CHAPTER 7 -Geometry of the Circle SIMSON LINE Definitions and propositions Solved problems RADICAL AXIS-COAXAL CIRCLES Definitions and propositions Solved problems POLES AND POLARS Definitions and propositions Solved problems SIMILITUDE AND INVERSION Definitions and propositions Solved problems Miscellaneous exercises CHAPTER 8 -Space Geometry Theorems and corollaries Solved problems Miscellaneous exercises INDEX PREFACE This book is intended as a second course in Euclidean geometry. Its purpose is to give the reader facility in applying the theorems of Euclid to the solution of geometrical problems. Each chapter begins with a brief account of Euclid's theorems and corollaries for simpli- city of reference, then states and proves a number of important propositions. Chapters close with a section of miscellaneous problems of increasing complexity, selected from an immense mass of material for their usefulness and interest. Hints of solutions for a large number of problems are also included. Since this is not intended for the beginner in geometry, such familiar concepts as point, line, ray, segment, angle, and polygon are used freely without explicit definition. For the purpose of clarity rather than rigor the general term line is used to designate sometimes a ray, sometimes a segment, sometimes the length of a segment; the meaning intended will be clear from the context. Definitions of less familiar geometrical concepts, such as those of modern and space geometry, are added for clarity, and since the use of symbols might prove an additional difficulty to some readers, geometrical notation is introduced gradually in each chapter. January, 1968 M. N. AREF WILLIAM WERNICK xi SYMBOLS EMPLOYED IN THIS BOOK L angle Q circle O parallelogram o quadrilateral 0 rectangle square 0 triangle a = ba equals b a > ba is greater than b a < ba is less than b a b a is parallel to b a b a is perpendicular to b a / b a divided by b a : bthe ratio of a to b AB2 the square of the distance from A to B because therefore xiii CHAPTER 1 TRIANGLES AND POLYGONS Theorems and Corollaries LINES AND ANGLES I.I. If a straight line meets another straight line, the sum of the two adjacent angles is two right angles. COROLLARY 1. If any number of straight lines are drawn from a given point, the sum of the consecutive angles so formed is, four right angles. COROLLARY 2. If a straight line meets another straight line, the bisectors of the two adjacent angles are at right angles to one another. 1.2. If the sum of two adjacent angles is two right angles, their non- coincident arms are in the same straight line. 1.3. If two straight lines intersect, the vertically opposite angles are equal. 1.4. If a straight line cuts two other straight lines so as to make the alternate angles equal, the two straight lines are parallel. 1.5. If a straight line cuts two other straight lines so as to make: (i) two corresponding angles equal; or (ii) the interior angles, on the same side of the line, supplementary, the two straight lines are parallel. 1.6. If a straight line intersects two parallel straight lines, it makes: (i) alternate angles equal; (ii) corresponding angles equal; (iii) two interior angles on the same side of the line supplementary. COROLLARY. Two angles whose respective arms are either parallel or perpendicular to one another are either equal or supplementary. 1.7. Straight lines which are parallel to the same straight line are parallel to one another. TRIANGLES AND THEIR CONGRUENCE 1.8. If one side of a triangle is produced, (i) the exterior angle is equal to the sum of the interior non-adjacent angles; (ii) the sum of the three angles of a triangle is two right angles. COROLLARY 1. If two angles of one triangle are respectively equal to two angles of another triangle, the third angles are equal and the triangles are equiangular. COROLLARY 2. If one side of a triangle is produced, the exterior angle is greater than either of the interior non-adjacent angles. 2 PROBLEMS AND SOLUTIONS IN EUCLIDEAN GEOMETRY COROLLARY 3. The sum of any two angles of a triangle is less than two right angles. 1.9. If all the sides of a polygon of n sides are produced in order, the sum of the exterior angles is four right angles. COROLLARY. The sum of all the interior angles of a polygon of n sides is (2n - 4) right angles. 1.10. Two triangles are congruent if two sides and the included angle of one triangle are respectively equal to two sides and the included angle of the other. 1.11. Two triangles are congruent if two angles and a side of one triangle are respectively equal to two angles and the corresponding side of the other. 1.12. If two sides of a triangle are equal, the angles opposite to these sides are equal. COROLLARY 1. The bisector of the vertex angle of an isosceles triangle, (i) bisects the base; (ii) is perpendicular to the base. COROLLARY 2. An equilaterial triangle is also equiangular. 1.13. If two angles of a triangle are equal, the sides which subtend these angles are equal. COROLLARY. An equiangular triangle is also equilateral. 1.14. Two triangles are congruent if the three sides of one triangle are respectively equal to the three sides of the other. 1.15. Two right-angled triangles are congruent if the hypotenuse and a side of one are respectively equal to the hypotenuse and a side of the other. INEQUALITIES 1.16. If two sides of a triangle are unequal, the greater side has the greater angle opposite to it. 1.17. If two angles of a triangle are unequal, the greater angle has the greater side opposite to it. 1.18. Any two sides of a triangle are together greater than the third. 1.19. If two triangles have two sides of the one respectively equal to two sides of the other and the included angles unequal, then the third side of that with the greater angle is greater than the third side of the other.