SOLUTION MANUAL For A SURVEY OF CLASSICAL AND
MODERN GEOMETRIES With Computer Activities 1st
Edition By Arthur Baragar
,https://www.stuvia.com/user/nursecare
SOLUTION MANUAL
For
A SURVEY OF
CLASSICAL AND MODERN
GEOMETRIES
With Computer Activities
1st Edition
By
Arthur Baragar
,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
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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 Poincar´e Models of Hyperbolic Geometry 149
7.1 The Poincar´e 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 Poincar´e 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
MODERN GEOMETRIES With Computer Activities 1st
Edition By Arthur Baragar
,https://www.stuvia.com/user/nursecare
SOLUTION MANUAL
For
A SURVEY OF
CLASSICAL AND MODERN
GEOMETRIES
With Computer Activities
1st Edition
By
Arthur Baragar
,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
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v
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 Poincar´e Models of Hyperbolic Geometry 149
7.1 The Poincar´e 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 Poincar´e 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
