Methods of Geometry

An Instructor's Manual presenting detailed solutions to all the problems in the book is available upon request from the Wiley editorial department.

JAMES T. SMITH, PhD, is Professor of Mathematics at San Francisco State University.

Preface

xi

About the author

xv

Introduction

1

(18)

Episodes

2

(7)

Advanced geometry

9

(4)

This book

13

(3)

Reading about geometry

16

(1)

Projects

17

(2)

Foundations

19

(34)

Geometry as applied mathematics

20

(3)

Need for rigor

23

(3)

Axiomatic method

26

(3)

Euclid's Elements

29

(4)

Coordinate geometry

33

(2)

Foundation problem

35

(2)

Parallel axiom

37

(2)

Firm foundations

39

(3)

Geometry as pure mathematics

42

(8)

Exercises and projects

50

(3)

Elementary Euclidean geometry

53

(74)

Incidence geometry

55

(2)

Ruler axiom and its consequences

57

(3)

Pasch's axiom and the separation theorems

60

(4)

Angles and the protractor axioms

64

(3)

Congruence

67

(4)

Perpendicularity

71

(5)

Parallel axiom and related theorems

76

(5)

Area and Pythagoras' theorem

81

(8)

Similarity

89

(3)

Polyhedral volume

92

(8)

Coordinate geometry

100

(6)

Circles and spheres

106

(4)

Arcs and trigonometric functions

110

(9)

π

119

(8)

Exercises on elementary geometry

127

(30)

Exercises on the incidence and ruler axioms

128

(2)

Exercises related to Pasch's axiom

130

(3)

Exercises on congruence and perpendicularity

133

(2)

Exercises involving the parallel axiom

135

(2)

Exercises on similarity and Pythagoras' theorem

137

(3)

Exercises on circles and spheres, part 1

140

(3)

Exercises on area

143

(3)

Exercises on volume

146

(2)

Exercises on circles and spheres, part 2

148

(3)

Exercises on coordinate geometry

151

(6)

Some triangle and circle geometry

157

(70)

Four concurrence theorems

158

(4)

Menelaus' theorem

162

(3)

Desargues' theorem

165

(6)

Ceva's theorem

171

(6)

Trigonometry

177

(9)

Vector products

186

(7)

Centroid

193

(3)

Orthocenter

196

(3)

Incenter and excenters

199

(5)

Euler line and Feuerbach circle

204

(7)

Exercises

211

(16)

Plane isometries and similarities

227

(68)

Transformations

232

(8)

Isometries

240

(7)

Reflections

247

(4)

Translations

251

(3)

Rotations

254

(4)

Structure theorem

258

(3)

Glide reflections

261

(1)

Isometries and orthogonal matrices

262

(5)

Classifying isometries

267

(9)

Similarities

276

(5)

Exercises

281

(14)

Three dimensional isometries and similarities

295

(32)

Isometries

296

(6)

Reflections

302

(4)

Translations and rotations

306

(3)

Glide and rotary reflections

309

(3)

Classifying isometries

312

(3)

Similarities

315

(2)

Exercises

317

(10)

Symmetry

327

(96)

Polygonal symmetry

336

(8)

Friezes

344

(11)

Wallpaper ornaments

355

(26)

Polyhedra

381

(26)

Exercises

407

(16)

Appendix A Equivalence relations

423

(2)

Appendix B Least upper bound principle

425

(4)

Appendix C Vector and matrix algebra

429

(14)

Bibliography

443

(20)

Index

463

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Summary

Author Biography

Table of Contents

Supplemental Materials