Geometry Illuminated: An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry

Contents ... 8
I Neutral Geometry ... 8
II Euclidean Geometry ... 9
III Euclidean Transformations ... 11
IV Hyperbolic Geometry ... 12
Preface ... 16
0 Axioms and Models ... 18
0.1 Fano\'s geometry ... 19
0.2 Further reading ... 21
0.3 Exercises ... 22
Part I Neutral Geometry ... 24
1 The Ax ioms of Incidence and Order ... 26
I.I Incidence ... 28
1.2 Order ... 29
1.3 Putting points in order ... 31
I. 4 Exercises ... 33
I .5 Further reading ... 34
2 Angles and Triangles ... 36
2. I Exercises ... 41
2 .2 References ... 42
3 Congruence Verse I: SAS and ASA ... 44
3.I Triangle congruence ... 47
3 .2 Exercises ... 51
4 Congruence Verse II: MS ... 54
4.I Supplementary angles ... 54
4.2 The alternate interior angle theorem ... 57
4.3 The exterior angle theorem ... 59
4.4 AAS ... 60
4.5 Exerci ses ... 61
5 Congruence Verse Ill: SSS ... 62
5.I Exercises ... 66
6 Distance, Length, and the Axioms of Continuity ... 68
6.I Synthetic comparison ... 68
6.2 Distance ... 70
6. 3 Exercises ... 79
7 Angle Measure ... 80
7.I Synthetic angle comparison ... 80
7.2 Right angles ... 84
7.3 Angle measure ... 87
7.4 Exercises ... 88
8 Triangles in Neutral Geometry ... 90
8.1 Exercises ... 95
8.2 References ... 96
9 Polygons ... 98
9.1 Definitions ... 98
9.2 Counting polygons ... 100
9.3 Interiors and exteriors ... 101
9.4 Interior angles: two dilemmas ... 105
9.5 Polygons of note ... 109
9.6 Exercises ... 110
10 Quadrilateral Congruence Theorems ... 112
I 0.1 Terminology ... 112
I 0.2 Quadrilateral congruence ... 113
I 0.2.1 SASAS, ASASA, and MSAS ... 114
10.2.2 SSSSA ... 117
I0.2.3 ASAAS, ASASS, ASSAS, MAAS,and SSSM ... 118
10.l.4 MASS ... 118
I 0.3 Exercises ... 120
Part II Euclidean Geometry ... 122
II The Axiom on Parallels ... 124
II .I Exercises ... 130
12 Parallel Projection ... 132
12.I Parallel projection ... 133
12.2 Parallel projection, order, and congruence ... 134
12.3 Parallel projection and distance ... 137
12. 4 Exercises ... 141
13 Similarity ... 142
I3. I Triangle similarity theorems ... 144
I3.2 The Pythagorean theorem ... 147
13.3 Exercises ... 150
14 Circles ... 152
14.I Definitions ... 152
14.2 Intersections ... 154
14.3 The inscribed angle theorem ... 158
14.4 Applications of the inscribed angle theorem ... 161
14.5 Exercises ... 164
14.6 References ... 165
15 Circumference ... 166
15.1 A theorem on perimeters ... 166
I5.2 Circumference ... 167
15.3 Lengths of arcs and radians ... 173
I5.4 Exercises ... 175
I5.5 References ... 175
16 Euclidean Constructions ... 176
17 Concurrence I ... 196
I7.I The circumcenter ... 196
I 7.2 The orthocenter ... 198
17.3 The centroid ... 201
I7.4 The incenter ... 203
I8 Concurrence II ... 208
I 8.I The Euler line ... 208
18.2 The nine point circle ... 210
18.3 The center of the nine point circle ... 213
18.4 Exercises ... 215
19 Concurrence Ill ... 216
19. I Excenters and excircles ... 216
19.2 Ceva\'s theorem ... 217
19.3 Menelaus\'s theorem ... 222
19.4 The Nagel point ... 224
19.5 Exercises ... 226
20 Trilinear Coordinates ... 228
20.1 Trilinear coordinates ... 228
20.2 Trilinears of the classical centers ... 232
20.3 Exercises ... 239
Part Ill Euclidean Transformation ... 240
21 Analytic Geometry ... 242
21.1 Analytic geometry ... 242
2I.2 The unit circle approach to trigonometry ... 247
21.3 Exercises ... 252
22 lsometries ... 256
22.1 Definitions ... 256
22.2 Fixed points ... 260
22.3 The analytic viewpoint ... 262
22.4 Exercises ... 264
23 Reflections ... 266
23.1 The analytic viewpoint ... 271
23.2 Exercises ... 273
24 Translations and Rotations ... 274
24.1 Translation ... 275
24.2 Rotations ... 278
24.3 The analytic viewpoint ... 280
24.4 Exercises ... 281
25 Orientation ... 284
25. I Exercises ... 288
26 Glide Reflections ... 290
26.I Glide reflections ... 290
26.2 Compositions of three reflections ... 292
26.3 Exercises ... 297
27 Change of Coordinates ... 298
27.I Vector arithmetic ... 298
27.2 Change of coordinates ... 302
27.3 Exercises ... 307
28 Dilation ... 308
28.I Similarity mappings ... 308
28.2 Dilations ... 310
28.3 Preserving incidence, order, and congruence ... 314
28.4 Exercises ... 317
29 Applications of Transformations ... 320
29.1 Varignon\'s theorem ... 320
29.2 Napoleon\'s theorem ... 322
29.3 The nine point circle ... 325
29.4 References ... 327
29.5 Exercises ... 328
30 Area I ... 330
30.1 The are a function ... 330
30.2 The laws of sines and cosines ... 336
30.3 Heron\'s formula ... 341
30.4 References ... 344
30.5 Exercises ... 344
31 Area II ... 346
3I.I Areas of polygons ... 346
3I.2 The area of a circle ... 353
31.3 Exercises ... 355
32 Barycentric Coordinates ... 358
32.I The vector approach ... 359
32.2 The connection to area and trilinears ... 363
32.3 Barycentric coordinates of triangle centers ... 367
32.4 References ... 369
32.5 Exercises ... 369
33 Inversion ... 370
33.I Stereographic projection ... 370
33.2 Inversion ... 375
33.3 Exercises ... 384
34 Inversion II ... 386
34.1 Complex numbers, complex arithmetic ... 386
34.1.1 Taylor series: a quick review ... 387
34.2 The geometry of complex arithmetic ... 390
34.3 Properties of the norm and conjugate ... 394
34.4 Exercises ... 395
35 Applications of Inversion ... 398
35.I Orthogonal circles ... 398
35.2 The arbelos ... 404
35.3 Steiner\'s porism ... 406
35.4 Exercises ... 408
Part IV Hyperbolic Geometry ... 412
36 The Search for a Rectangle ... 414
36.1 If there were a rectangle ... ... 414
36.2 The search for a rectangle ... 420
36.3 References ... 426
37 Non-Euclidean Parallels ... 428
37.I Exercises ... 436
38 The Pseudosphere ... 438
38.1 Surfaces ... 439
38.2 Maps between surfaces. The Gauss map ... 442
38.3 Gaussian curvature ... 445
38.4 The tractrix and pseudosphere ... 447
38.5 Exercises ... 449
39 Geodesics on the Pseudosphere ... 450
39.1 Geodesics, the theory ... 450
39.2 Geodesics, the calculations ... 451
39.3 References ... 459
39.4 Exercises ... 459
40 The Upper Half Plane ... 460
40.I Distance ... 460
40.2 Angle measure ... 463
40.3 Extending the domain ... 467
40.4 Exercises ... 469
41 The Poincare disk ... 472
41.1 The Poincare disk model ... 472
41.2 Interpreting \"undefineds\" in the Poincare disk ... 474
41.3 Verify ing the axioms ... 475
4I.4 Exercises ... 490
42 Hyperbolic Reflections ... 492
42.I Exercises ... 499
43 Orientation-Preserving Hyperbolic lsometries ... 500
43.1 An important example ... 502
43.2 Classification by fixed points ... 503
43.3 References ... 507
43.4 Exercises ... 507
44 The Six Hyperbolic Trigonometric Functions ... 510
45 Hyperbolic Trigonometry ... 518
45.1 Pythagorean theorem ... 519
45.2 Sine and cosine in a hyperbolic triangle ... 522
45.3 Circumference of a hyperbolic circle ... 526
45.4 On a small scale ... 528
45.5 Exercises ... 529
46 Hyperbolic Area ... 532
46.I Area on the pseudosphere ... 533
46.2 Areas of polygons in the Poincare disk ... 539
46.3 Area of a circle ... 542
46.4 Exercises ... 544
47 Tiling ... 546
47.I Exercises ... 552
Bibliography ... 554
Index ... 556