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
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
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 Con struction 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 Isometries ................ 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 Isometries .................................. 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
9 The Product of Direct Isometries ........................... 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 III 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
14 Reciprocation and the Extended Plane ....................... 375
14.1 Harmonic Conjugates ................................... 375
14.2 The Projective Plane and Reciprocation ................ 385
14.3 Conjugate Points and Lines ............................ 396
14.4 Conies ................................................ 402
14.5 Problems .............................................. 409
15 Cross Ratios ............................................... 411
15.1 Cross Ratios .......................................... 411
15.2 Applications of Cross Ratios .......................... 422
15.3 Problems .............................................. 431
16 Introduction to Projective Geometry ........................ 435
16.1 Straightedge Constructions ............................ 435
16.2 Perspectivities and Projectivities .................... 445
16.3 Line Perspectivities and Line Projectivities .......... 450
16.4 Projective Geometry and Fixed Points .................. 450
16.5 Projecting a Line to Infinity ......................... 453
16.6 The Apollonian Definition of a Conic .................. 457
16.7 Problems .............................................. 463
Bibliography .................................................. 466
Index ......................................................... 471
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