CLASSICAL GEOMETRY CLASSICAL GEOMETRY Euclidean, Transformational, Inversive, and Projective I. E. Leonard, J. E. Lewis, A. C. F. Liu, G. W. Tokarsky Department of Mathematical and Statistical Sciences University of Alberta WILEY Copyright© 2014 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission ofthe Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. 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Lewis, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada, A. C. F. Liu, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada, G. W. Tokarsky, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada. pages em Includes bibliographical references and index. ISBN 978-1-118-67919-7 (hardback) 1. Geometry. I. Lewis, J. E. (James Edward) author. II. Liu, A. C. F. (Andrew Chiang-Fung) author. III. Tokarsky, G. W., author. IV. Title. QA445.L46 2014 516-dc23 2013042035 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1 CONTENTS Preface xi PART I EUCLIDEAN GEOMETRY 1 Congruency 3 1.1 Introduction 3 1.2 Congruent Figures 6 1.3 Parallel Lines 12 1.3.1 Angles in a Triangle 13 1.3.2 Thales' Theorem 14 1.3.3 Quadrilaterals 17 1.4 More About Congruency 21 1.5 Perpendiculars and Angle Bisectors 24 1.6 Construction Problems 28 1.6.1 The Method of Loci 31 1.7 Solutions to Selected Exercises 33 1.8 Problems 38 v vi CONTENTS 2 Concurrency 41 2.1 Perpendicular Bisectors 41 2.2 Angle Bisectors 43 2.3 Altitudes 46 2.4 Medians 48 2.5 Construction Problems 50 2.6 Solutions to the Exercises 54 2.7 Problems 56 3 Similarity 59 3.1 Similar Triangles 59 3.2 Parallel Lines and Similarity 60 3.3 Other Conditions Implying Similarity 64 3.4 Examples 67 3.5 Construction Problems 75 3.6 The Power of a Point 82 3.7 Solutions to the Exercises 87 3.8 Problems 90 4 Theorems of Ceva and Menelaus 95 4.1 Directed Distances, Directed Ratios 95 4.2 The Theorems 97 4.3 Applications of Ceva's Theorem 99 4.4 Applications of Menelaus' Theorem 103 4.5 Proofs of the Theorems 115 4.6 Extended Versions of the Theorems 125 4.6.1 Ceva's Theorem in the Extended Plane 127 4.6.2 Menelaus' Theorem in the Extended Plane 129 4.7 Problems 131 5 Area 133 5.1 Basic Properties 133 5.1.1 Areas of Polygons 134 5.1.2 Finding the Area of Polygons 138 5.1.3 Areas of Other Shapes 139 5.2 Applications of the Basic Properties 140 CONTENTS vii 5.3 Other Formulae for the Area of a Triangle 147 5.4 Solutions to the Exercises 153 5.5 Problems 153 6 Miscellaneous Topics 159 6.1 The Three Problems of Antiquity 159 6.2 Constructing Segments of Specific Lengths 161 6.3 Construction of Regular Polygons 166 6.3.1 Construction of the Regular Pentagon 168 6.3.2 Construction of Other Regular Polygons 169 6.4 Miquel's Theorem 171 6.5 Morley's Theorem 178 6.6 The Nine-Point Circle 185 6.6.1 Special Cases 188 6.7 The Steiner-Lehmus Theorem 193 6.8 The Circle of Apollonius 197 6.9 Solutions to the Exercises 200 6.10 Problems 201 PART II TRANSFORMATIONAL GEOMETRY 7 The Euclidean Transformations or lsometries 207 7.1 Rotations, Reflections, and Translations 207 7.2 Mappings and Transformations 211 7.2.1 Isometries 213 7.3 Using Rotations, Reflections, and Translations 217 7.4 Problems 227 8 The Algebra of lsometries 235 8.1 Basic Algebraic Properties 235 8.2 Groups of Isometries 240 8.2.1 Direct and Opposite Isometries 241 8.3 The Product of Reflections 245 8.4 Problems 250 viii CONTENTS 9 The Product of Direct lsometries 255 9.1 Angles 255 9.2 Fixed Points 257 9.3 The Product of Two Translations 258 9.4 The Product of a Translation and a Rotation 259 9.5 The Product of Two Rotations 262 9.6 Problems 265 10 Symmetry and Groups 271 10.1 More About Groups 271 10.1.1 Cyclic and Dihedral Groups 275 10.2 Leonardo's Theorem 279 10.3 Problems 283 11 Homotheties 289 11.1 The Pantograph 289 11.2 Some Basic Properties 290 11.2.1 Circles 293 11.3 Construction Problems 295 11.4 Using Homotheties in Proofs 300 11.5 Dilatation 304 11.6 Problems 306 12 Tessellations 313 12.1 Tilings 313 12.2 Monohedral Tilings 314 12.3 Tiling with Regular Polygons 319 12.4 Platonic and Archimedean Tilings 325 12.5 Problems 332 PART Ill INVERSIVE AND PROJECTIVE GEOMETRIES 13 Introduction to Inversive Geometry 339 13.1 Inversion in the Euclidean Plane 339 13.2 The Effect of Inversion on Euclidean Properties 345 13.3 Orthogonal Circles 353 13.4 Compass-Only Constructions 362 13.5 Problems 371