Classical Geometry - Euclidean, Transformational, Inversive, and Projective

493 Pages·2014·30.54 MB·English

by  I. E. Leonard

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
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Library of Congress Cataloging-in-Publication Data:
Leonard, I. Ed., 1938-author.
Classical geometry : Euclidean, transformational, inversive, and projective I I. E. Leonard,
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada,
J. E. 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