SOLUTION MANUAL l
For
ASURVEYOFCLASSICAL
L L L
AND MODERN GEOMETRIES
L L
With Computer Activities
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1stEdition
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By
Arthur Baragar
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,Contents
1 Euclidean Geometry l 1
1.1 The Pythagorean Theorem ................................................................3
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1.2 The Axioms of Euclidean Geometry .................................................5
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1.3 SSS, SAS, and ASA ............................................................................7
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1.4 Parallel Lines ..................................................................................... 11
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1.5 Pons Asinorum .................................................................................. 12
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1.6 The Star Trek Lemma ..................................................................... 12
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1.7 Similar Triangles ............................................................................... 18
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1.8 Power of the Point ............................................................................. 24
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1.9 The Medians and Centroid.............................................................. 33
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1.10 The Incircle, Excircles, and the Law of Cosines........................... 35
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1.11 The Circumcircle and Law of Sines ............................................... 42
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1.12 The Euler Line .................................................................................. 48
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1.13 The Nine Point Circle ...................................................................... 50
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1.14 Pedal Triangles and the Simson Line ............................................. 57
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1.15 Menelaus and Ceva............................................................................ 67
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2 Geometry in Greek Astronomy l l 75 l
2.1 The Relative Size of the Moon and Sun........................................ 75
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2.2 The Diameter of the Earth .............................................................. 76
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3 Constructions Using a Compass and Straightedge l 81 l l l l
3.1 The Rules .......................................................................................... 81
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3.2 Some Examples .................................................................................. 81
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3.3 Basic Results ..................................................................................... 82
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3.4 The Algebra of Constructible Lengths .......................................... 92
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3.5 The Regular Pentagon ..................................................................... 94
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3.6 Other Constructible Figures ......................................................... 102
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3.7 Trisecting an Arbitrary Angle....................................................... 105
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4 Geometer’s Sketchpad l 111
4.1 The Rules of Constructions ........................................................... 111
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4.2 Lemmas and Theorems .................................................................. 111
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4.3 Archimedes’ Trisection Algorithm ................................................. 114
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, 4.4 Verification of Theorems .................................................................114
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4.5 Sophisticated Results ......................................................................117 l
4.6 Parabola Paper ................................................................................120
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5 Higher Dimensional Objects
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5.1 The Platonic Solids ........................................................................125
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5.2 The Duality of Platonic Solids .....................................................127
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5.3 The Euler Characteristic.................................................................127
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5.4 Semiregular Polyhedra ....................................................................127
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5.5 A Partial Categorization of Semiregular Polyhedra ...................130
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5.6 Four-Dimensional Objects ...............................................................138 l
6 Hyperbolic Geometry l 143
6.1 Models ...............................................................................................143
6.2 Results from Neutral Geometry.....................................................143
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6.3 The Congruence of Similar Triangles ...........................................145
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6.4 Parallel and Ultraparallel Lines .....................................................145
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6.5 Singly Asymptotic Triangles .........................................................146
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6.6 Doubly and Triply Asymptotic Triangles....................................146
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6.7 The Area of Asymptotic Triangles ...............................................147
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7 The lPoincaré Models of Hyperbolic Geometry l 149 l l l
7.1 The Poincar´e Upper Half Plane Model .........................................149
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7.2 Vertical (Euclidean) Lines ..............................................................149
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7.3 Isometries .........................................................................................149
7.4 Inversion in the Circle .....................................................................150
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7.5 Inversion in Euclidean Geometry ................................................... 161
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7.6 Fractional Linear Transformations ................................................164
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7.7 The Cross Ratio ...............................................................................169
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7.8 Translations ......................................................................................173
7.9 Rotations ...........................................................................................177
7.10 Reflections ........................................................................................181
7.11 Lengths..............................................................................................185
7.12 The Axioms of Hyperbolic Geometry ...........................................186
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7.13 The Area of Triangles.....................................................................186
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7.14 The Poincar´e Disc Model ................................................................188
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7.15 Circles and Horocycles ....................................................................190
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7.16 Hyperbolic Trigonometry ...............................................................195
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7.17 The Angle of Parallelism ................................................................207
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7.18 Curvature ..........................................................................................209
8 Tilings and Lattices l l 211
8.1 Regular Tilings .................................................................................211
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8.2 Semiregular Tilings ..........................................................................211
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8.3 Lattices and Fundamental Domains ..............................................212
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8.4 Tilings in Hyperbolic Space ...........................................................212
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8.5 Tilings in Art ................................................................................... 220
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, 9 Foundations 221
9.1 Theories ........................................................................................... 221
9.2 The Real Line ................................................................................. 221
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9.3 The Plane ........................................................................................ 221
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9.4 Line Segments and Lines ................................................................ 221
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9.5 Separation Axioms .......................................................................... 222
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9.6 Circles ............................................................................................... 225
9.7 Isometries and Congruence ............................................................ 226
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9.8 The Parallel Postulate .................................................................... 227
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9.9 Similar Triangles ............................................................................. 227
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10 Spherical Geometry l 229
10.1 The Area of Triangles .................................................................... 229
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10.2 The Geometry of Right Triangles ................................................ 231
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10.3 The Geometry of Spherical Triangles .......................................... 232
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10.4 Menelaus’ Theorem ......................................................................... 234
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10.5 Heron’s Formula .............................................................................. 241
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10.6 Tilings of the Sphere ...................................................................... 245
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10.7 The Axioms ..................................................................................... 247
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10.8 Elliptic Geometry ........................................................................... 247
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11 Projective Geometry l 249
11.1 Moving a Line to Infinity .............................................................. 249
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11.2 Pascal’s Theorem ............................................................................ 250
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11.3 Projective Coordinates ................................................................... 250
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11.4 Duality .............................................................................................. 255
11.5 Dual Conics and Brianchon’s Theorem ....................................... 257
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11.6 Areal Coordinates ........................................................................... 258
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12 The l Pseudosphere in Lorentz Space 265 l l l
12.1 The Sphere as a Foil....................................................................... 265
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12.2 The Pseudosphere .......................................................................... 272
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12.3 Angles and the Lorentz Cross Product ........................................ 280
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12.4 A Different Perspective .................................................................. 284
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12.5 The Beltrami-Klein Model............................................................. 286
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12.6 Menelaus’ Theorem ......................................................................... 286
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For
ASURVEYOFCLASSICAL
L L L
AND MODERN GEOMETRIES
L L
With Computer Activities
l l
1stEdition
l
By
Arthur Baragar
l
,Contents
1 Euclidean Geometry l 1
1.1 The Pythagorean Theorem ................................................................3
l l
1.2 The Axioms of Euclidean Geometry .................................................5
l l l l
1.3 SSS, SAS, and ASA ............................................................................7
l l l
1.4 Parallel Lines ..................................................................................... 11
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1.5 Pons Asinorum .................................................................................. 12
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1.6 The Star Trek Lemma ..................................................................... 12
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1.7 Similar Triangles ............................................................................... 18
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1.8 Power of the Point ............................................................................. 24
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1.9 The Medians and Centroid.............................................................. 33
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1.10 The Incircle, Excircles, and the Law of Cosines........................... 35
l l l l l l l
1.11 The Circumcircle and Law of Sines ............................................... 42
l l l l l
1.12 The Euler Line .................................................................................. 48
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1.13 The Nine Point Circle ...................................................................... 50
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1.14 Pedal Triangles and the Simson Line ............................................. 57
l l l l l
1.15 Menelaus and Ceva............................................................................ 67
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2 Geometry in Greek Astronomy l l 75 l
2.1 The Relative Size of the Moon and Sun........................................ 75
l l l l l l l
2.2 The Diameter of the Earth .............................................................. 76
l l l l
3 Constructions Using a Compass and Straightedge l 81 l l l l
3.1 The Rules .......................................................................................... 81
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3.2 Some Examples .................................................................................. 81
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3.3 Basic Results ..................................................................................... 82
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3.4 The Algebra of Constructible Lengths .......................................... 92
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3.5 The Regular Pentagon ..................................................................... 94
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3.6 Other Constructible Figures ......................................................... 102
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3.7 Trisecting an Arbitrary Angle....................................................... 105
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4 Geometer’s Sketchpad l 111
4.1 The Rules of Constructions ........................................................... 111
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4.2 Lemmas and Theorems .................................................................. 111
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4.3 Archimedes’ Trisection Algorithm ................................................. 114
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v
, 4.4 Verification of Theorems .................................................................114
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4.5 Sophisticated Results ......................................................................117 l
4.6 Parabola Paper ................................................................................120
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5 Higher Dimensional Objects
l 125 l
5.1 The Platonic Solids ........................................................................125
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5.2 The Duality of Platonic Solids .....................................................127
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5.3 The Euler Characteristic.................................................................127
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5.4 Semiregular Polyhedra ....................................................................127
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5.5 A Partial Categorization of Semiregular Polyhedra ...................130
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5.6 Four-Dimensional Objects ...............................................................138 l
6 Hyperbolic Geometry l 143
6.1 Models ...............................................................................................143
6.2 Results from Neutral Geometry.....................................................143
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6.3 The Congruence of Similar Triangles ...........................................145
l l l l
6.4 Parallel and Ultraparallel Lines .....................................................145
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6.5 Singly Asymptotic Triangles .........................................................146
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6.6 Doubly and Triply Asymptotic Triangles....................................146
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6.7 The Area of Asymptotic Triangles ...............................................147
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7 The lPoincaré Models of Hyperbolic Geometry l 149 l l l
7.1 The Poincar´e Upper Half Plane Model .........................................149
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7.2 Vertical (Euclidean) Lines ..............................................................149
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7.3 Isometries .........................................................................................149
7.4 Inversion in the Circle .....................................................................150
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7.5 Inversion in Euclidean Geometry ................................................... 161
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7.6 Fractional Linear Transformations ................................................164
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7.7 The Cross Ratio ...............................................................................169
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7.8 Translations ......................................................................................173
7.9 Rotations ...........................................................................................177
7.10 Reflections ........................................................................................181
7.11 Lengths..............................................................................................185
7.12 The Axioms of Hyperbolic Geometry ...........................................186
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7.13 The Area of Triangles.....................................................................186
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7.14 The Poincar´e Disc Model ................................................................188
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7.15 Circles and Horocycles ....................................................................190
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7.16 Hyperbolic Trigonometry ...............................................................195
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7.17 The Angle of Parallelism ................................................................207
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7.18 Curvature ..........................................................................................209
8 Tilings and Lattices l l 211
8.1 Regular Tilings .................................................................................211
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8.2 Semiregular Tilings ..........................................................................211
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8.3 Lattices and Fundamental Domains ..............................................212
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8.4 Tilings in Hyperbolic Space ...........................................................212
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8.5 Tilings in Art ................................................................................... 220
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, 9 Foundations 221
9.1 Theories ........................................................................................... 221
9.2 The Real Line ................................................................................. 221
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9.3 The Plane ........................................................................................ 221
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9.4 Line Segments and Lines ................................................................ 221
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9.5 Separation Axioms .......................................................................... 222
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9.6 Circles ............................................................................................... 225
9.7 Isometries and Congruence ............................................................ 226
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9.8 The Parallel Postulate .................................................................... 227
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9.9 Similar Triangles ............................................................................. 227
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10 Spherical Geometry l 229
10.1 The Area of Triangles .................................................................... 229
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10.2 The Geometry of Right Triangles ................................................ 231
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10.3 The Geometry of Spherical Triangles .......................................... 232
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10.4 Menelaus’ Theorem ......................................................................... 234
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10.5 Heron’s Formula .............................................................................. 241
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10.6 Tilings of the Sphere ...................................................................... 245
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10.7 The Axioms ..................................................................................... 247
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10.8 Elliptic Geometry ........................................................................... 247
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11 Projective Geometry l 249
11.1 Moving a Line to Infinity .............................................................. 249
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11.2 Pascal’s Theorem ............................................................................ 250
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11.3 Projective Coordinates ................................................................... 250
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11.4 Duality .............................................................................................. 255
11.5 Dual Conics and Brianchon’s Theorem ....................................... 257
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11.6 Areal Coordinates ........................................................................... 258
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12 The l Pseudosphere in Lorentz Space 265 l l l
12.1 The Sphere as a Foil....................................................................... 265
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12.2 The Pseudosphere .......................................................................... 272
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12.3 Angles and the Lorentz Cross Product ........................................ 280
l l l l l
12.4 A Different Perspective .................................................................. 284
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12.5 The Beltrami-Klein Model............................................................. 286
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12.6 Menelaus’ Theorem ......................................................................... 286
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