A SURVEY OF CLASSICAL

A SURVEY OF CLASSICAL

AND MODERN GEOMETRIES

With Computer Activities

SOLUTION MANUAL

For Arthur Baragar 1st Edition By 1 / 4

Contents

• Euclidean Geometry1

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 PonsAsinorum ......................... 12 1.6 TheStarTrekLemma ..................... 12 1.7 Similar Triangles........................ 18 1.8 PowerofthePoint ....................... 24 1.9 TheMediansandCentroid .................. 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

• Geometry in Greek Astronomy75

2.1 The Relative Size of the Moon and Sun........... 75 2.2 The Diameter of the Earth.................. 76

• Constructions Using a Compass and Straightedge 81

3.1 TheRules............................ 81 3.2 SomeExamples......................... 81 3.3 BasicResults .......................... 82 3.4 TheAlgebraofConstructibleLengths ............ 92 3.5 The Regular Pentagon..................... 94 3.6 OtherConstructibleFigures.................. 102 3.7 TrisectinganArbitraryAngle................. 105

• Geometer’s Sketchpad111

4.1 The Rules of Constructions.................. 111 4.2 Lemmas and Theorems..................... 111 4.3 Archimedes’ Trisection Algorithm............... 114 v 2 / 4

viContents 4.4 Verification of Theorems.................... 114 4.5 SophisticatedResults...................... 117 4.6 ParabolaPaper......................... 120

• Higher Dimensional Objects 125

5.1 The Platonic Solids....................... 125 5.2 The Duality of Platonic Solids................ 127 5.3 TheEulerCharacteristic.................... 127 5.4 Semiregular Polyhedra..................... 127 5.5 A Partial Categorization of Semiregular Polyhedra..... 130 5.6 Four-DimensionalObjects................... 138

• Hyperbolic Geometry 143

6.1 Models.............................. 143 6.2 ResultsfromNeutralGeometry................ 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

• The Poincar´e Models of Hyperbolic Geometry 149

7.1 The Poincar´eUpperHalfPlaneModel............ 149 7.2 Vertical(Euclidean)Lines................... 149 7.3 Isometries............................ 149 7.4 InversionintheCircle ..................... 150 7.5 InversioninEuclideanGeometry ............... 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 Poincar´eDiscModel ................... 188 7.15 Circles and Horocycles..................... 190 7.16 Hyperbolic Trigonometry................... 195 7.17 The Angle of Parallelism.................... 207 7.18 Curvature............................ 209

• 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 3 / 4

Contentsvii 8.5 Tilings in Art .......................... 220

• Foundations 221

9.1 Theories............................. 221 9.2 TheRealLine.......................... 221 9.3 ThePlane............................ 221 9.4 LineSegmentsandLines.................... 221 9.5 Separation Axioms....................... 222 9.6 Circles.............................. 225 9.7 Isometries and Congruence.................. 226 9.8 The Parallel Postulate..................... 227 9.9 Similar Triangles........................ 227 10 Spherical Geometry 229 10.1 The Area of Triangles..................... 229 10.2 The Geometry of Right Triangles............... 231 10.3 The Geometry of Spherical Triangles............. 232 10.4 Menelaus’ Theorem....................... 234 10.5 Heron’s Formula........................ 241 10.6 Tilings of the Sphere...................... 245 10.7 The Axioms . .......................... 247 10.8 Elliptic Geometry........................ 247 11 Projective Geometry 249 11.1 Moving a Line to Infinity................... 249 11.2 Pascal’s Theorem........................ 250 11.3 Projective Coordinates..................... 250 11.4 Duality............................. 255 11.5 Dual Conics and Brianchon’s Theorem............ 257 11.6 Areal Coordinates....................... 258 12 The Pseudosphere in Lorentz Space 265 12.1 The Sphere as a Foil...................... 265 12.2 The Pseudosphere....................... 272 12.3 Angles and the Lorentz Cross Product............ 280 12.4 A Different Perspective.................... 284 12.5 The Beltrami-Klein Model................... 286 12.6 Menelaus’ Theorem....................... 286

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