Methods of Geometry, Second Edition (Wiley-Interscience)

Cover......Page 1
Title......Page 2
Contents......Page 6
Preface......Page 10
About the author......Page 14
1 Introduction......Page 16
1.1 Episodes......Page 17
1.2 Advanced geometry......Page 24
1.3 This book......Page 28
1.4 Reading about geometry......Page 31
1.5 Projects......Page 32
2 Foundations......Page 34
2.1 Geometry as applied mathematics......Page 35
2.2 Need for rigor......Page 38
2.3 Axiomatic method......Page 41
2.4 Euclid\'s Elements......Page 44
2.5 Coordinate geometry......Page 48
2.6 Foundation problem......Page 50
2.7 Parallel axiom......Page 52
2.8 Firm foundations......Page 54
2.9 Geometry as pure mathematics......Page 57
2.10 Exercises and projects......Page 65
3 Elementary Euclidean geometry......Page 68
3.1 Incidence geometry......Page 70
3.2 Ruler axiom and its consequences......Page 72
3.3 Pasch\'s axiom and the separation theorems......Page 75
3.4 Angles and the protractor axioms......Page 79
3.5 Congruence......Page 82
3.6 Perpendicularity......Page 86
3.7 Parallel axiom and related theorems......Page 91
3.8 Area and Pythagoras\' theorem......Page 96
3.9 Similarity......Page 104
3.10 Polyhedral volume......Page 107
3.11 Coordinate geometry......Page 115
3.12 Circles and spheres......Page 121
3.13 Arcs and trigonometric functions......Page 125
3.14 Pi......Page 134
4 Exercises on elementary geometry......Page 142
4.1 Exercises on the incidence and ruler axioms......Page 143
4.2 Exercises related to Pasch\'s axiom......Page 145
4.3 Exercises on congruence and perpendicularity......Page 148
4.4 Exercises involving the parallel axiom......Page 150
4.5 Exercises on similarity and Pythagoras\' theorem......Page 152
4.6 Exercises on circles and spheres, part 1......Page 155
4.7 Exercises on area......Page 158
4.8 Exercises on volume......Page 161
4.9 Exercises on circles and spheres, part 2......Page 163
4.10 Exercises on coordinate geometry......Page 166
5 Some triangle and circle geometry......Page 172
5.1 Four concurrence theorems......Page 173
5.2 Menelaus\' theorem......Page 177
5.3 Desargues\' theorem......Page 180
5.4 Ceva\'s theorem......Page 186
5.5 Trigonometry......Page 192
5.6 Vector products......Page 201
5.7 Centroid......Page 208
5.8 Orthocenter......Page 211
5.9 Incenter and excenters......Page 214
5.10 Euler line and Feuerbach circle......Page 219
5.11 Exercises......Page 226
6 Plane isometrles and similarities......Page 242
6.1 Transformations......Page 247
6.2 Isometries......Page 255
6.3 Reflections......Page 262
6.4 Translations......Page 266
6.5 Rotations......Page 269
6.6 Structure theorem......Page 273
6.7 Glide reflections......Page 276
6.8 Isometries and orthogonal matrices......Page 277
6.9 Classifying isometries......Page 282
6.10 Similarities......Page 291
6.11 Exercises......Page 296
7 Three dimensional isometries and similarities......Page 310
7.1 Isometries......Page 311
7.2 Reflections......Page 317
7.3 Translations and rotations......Page 321
7.4 Glide and rotary reflections......Page 324
7.5 Classifying isometries......Page 327
7.6 Similarities......Page 330
7.7 Exercises......Page 332
8 Symmetry......Page 342
8.1 Polygonal symmetry......Page 351
8.2 Friezes......Page 359
8.3 Wallpaper ornaments......Page 370
8.4 Polyhedra......Page 396
8.5 Exercises......Page 422
Appendix A Equivalence relations......Page 438
Appendix B Least upper bound principle......Page 440
Appendix C Vector and matrix algebra......Page 444
Bibliography......Page 458
Index......Page 478