, A SURVEY OF CLASSICAL AND MODERN
GEOMETRIES WITH COMPUTER ACTIVITIES
1ST EDITION BY ARTHUR BARAGAR SOLUTION
MANUAL
Contents
1 Euclidean Geometry 1
1.1 The Pythagorean Theorem ................................................................ 3
1.2 The Axioms of Euclidean Geometry ................................................. 5
1.3 SSS, SAS, and ASA ........................................................................ 7
1.4 Parallel Lines ..................................................................................... 11
1.5 Pons Asinorum .................................................................................. 12
1.6 The Star Trek Lemma ...................................................................... 12
1.7 Similar Triangles ............................................................................... 18
1.8 Power of the Point ........................................................................... 24
1.9 The Medians and Centroid .............................................................. 33
1.10 The Incircle, Excircles, and the Law of Cosines ............................ 35
1.11 The Circumcircle and Law of Sines ................................................ 42
1.12 The Euler Line .................................................................................. 48
1.13 The Nine Point Circle ..................................................................... 50
1.14 Pedal Triangles and the Simson Line ............................................. 57
1.15 Menelaus and Ceva ........................................................................... 67
2 Geometry in Greek Astronomy 75
2.1 The Relative Size of the Moon and Sun ......................................... 75
2.2 The Diameter of the Earth .............................................................. 76
3 Constructions Using a Compass and Straightedge 81
3.1 The Rules ........................................................................................... 81
3.2 Some Examples ................................................................................. 81
3.3 Basic Results ..................................................................................... 82
3.4 The Algebra of Constructible Lengths ........................................... 92
3.5 The Regular Pentagon ...................................................................... 94
3.6 Other Constructible Figures .......................................................... 102
, 3.7 Trisecting an Arbitrary Angle ....................................................... 105
4 Geometer’s Sketchpad 111
4.1 The Rules of Constructions ........................................................... 111
4.2 Lemmas and Theorems .................................................................. 111
4.3 Archimedes’ Trisection Algorithm ................................................ 114
v
vi CONTENTS
4.4 Verification of Theorems ............................................................... 114
4.5 Sophisticated Results ..................................................................... 117
4.6 Parabola Paper ............................................................................... 120
5 Higher Dimensional Objects 125
5.1 The Platonic Solids ......................................................................... 125
5.2 The Duality of Platonic Solids .................................................... 127
5.3 The Euler Characteristic ................................................................ 127
5.4 Semiregular Polyhedra .................................................................. 127
5.5 A Partial Categorization of Semiregular Polyhedra ................... 130
5.6 Four-Dimensional Objects ............................................................. 138
6 Hyperbolic Geometry 143
6.1 Models ............................................................................................. 143
6.2 Results from Neutral Geometry .................................................... 143
6.3 The Congruence of Similar Triangles........................................... 145
6.4 Parallel and Ultraparallel Lines .................................................... 145
6.5 Singly Asymptotic Triangles.......................................................... 146
6.6 Doubly and Triply Asymptotic Triangles .................................... 146
6.7 The Area of Asymptotic Triangles ............................................... 147
7 The Poincaré Models of Hyperbolic Geometry 149
7.1 The Poincaré Upper Half Plane Model ........................................ 149
7.2 Vertical (Euclidean) Lines ............................................................. 149
7.3 Isometries ........................................................................................ 149
7.4 Inversion in the Circle ................................................................... 150
7.5 Inversion in Euclidean Geometry ................................................. 161
7.6 Fractional Linear Transformations .............................................. 164
7.7 The Cross Ratio .............................................................................. 169
7.8 Translations .................................................................................... 173
7.9 Rotations ......................................................................................... 177
7.10 Reflections ....................................................................................... 181
7.11 Lengths ............................................................................................ 185
7.12 The Axioms of Hyperbolic Geometry ........................................... 186
7.13 The Area of Triangles .................................................................... 186
7.14 The Poincaré Disc Model .............................................................. 188
7.15 Circles and Horocycles................................................................... 190
7.16 Hyperbolic Trigonometry .............................................................. 195
7.17 The Angle of Parallelism ............................................................... 207
7.18 Curvature......................................................................................... 209
, 8 Tilings and Lattices 211
8.1 Regular Tilings ................................................................................ 211
8.2 Semiregular Tilings ........................................................................ 211
8.3 Lattices and Fundamental Domains ............................................. 212
8.4 Tilings in Hyperbolic Space .......................................................... 212
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