Solution Manual For A Survey Of Classical
And Modern Geometries With Computer
Activities 1st Edition By Arthur.
, A SURVEY OF CLASSICAL AND MODERN
GEOMETRIESWITH COMPUTER ACTIVITIES
1STEDITIONBYARTHURBARAGARSOLUTION
MANUAL
Contents
1 EuclideanGeometry 1
ThePythagoreanTheorem ............................................................................... 3
TheAxiomsofEuclideanGeometry ................................................................. 5
SSS,SAS,andASA .............................................................................................. 7
ParallelLines ..................................................................................................... 11
PonsAsinorum.................................................................................................. 12
TheStarTrekLemma ....................................................................................... 12
SimilarTriangles .............................................................................................. 18
PowerofthePoint ............................................................................................. 24
TheMediansandCentroid .............................................................................. 33
TheIncircle,Excircles,andtheLawofCosines............................................ 35
TheCircumcircleandLawofSines ................................................................ 42
TheEulerLine .................................................................................................... 48
TheNinePointCircle ........................................................................................ 50
PedalTrianglesandtheSimsonLine............................................................. 57
MenelausandCeva ........................................................................................... 67
2 GeometryinGreekAstronomy 75
TheRelativeSizeoftheMoonandSun ........................................................... 75
TheDiameteroftheEarth................................................................................ 76
3 ConstructionsUsingaCompassandStraightedge 81
TheRules ............................................................................................................ 81
SomeExamples ................................................................................................. 81
BasicResults ...................................................................................................... 82
TheAlgebraofConstructibleLengths.......................................................... 92
TheRegularPentagon ..................................................................................... 94
OtherConstructibleFigures ........................................................................ 102
, TrisectinganArbitraryAngle ...................................................................... 105
4 Geometer’sSketchpad 111
TheRulesofConstructions ........................................................................... 111
LemmasandTheorems................................................................................. 111
Archimedes’TrisectionAlgorithm ............................................................ 114
v
vi CONTENTS
VerificationofTheorems .............................................................................. 114
SophisticatedResults.................................................................................... 117
ParabolaPaper................................................................................................ 120
5 HigherDimensionalObjects 125
ThePlatonicSolids ......................................................................................... 125
TheDualityofPlatonicSolids .......................................................................127
TheEulerCharacteristic ............................................................................... 127
SemiregularPolyhedra................................................................................. 127
APartialCategorizationofSemiregularPolyhedra................................ 130
Four-DimensionalObjects .......................................................................... 138
6 HyperbolicGeometry 143
Models............................................................................................................... 143
ResultsfromNeutralGeometry .................................................................. 143
TheCongruenceofSimilarTriangles ......................................................... 145
ParallelandUltraparallelLines................................................................... 145
SinglyAsymptoticTriangles........................................................................ 146
DoublyandTriplyAsymptoticTriangles .................................................. 146
TheAreaofAsymptoticTriangles ............................................................... 147
7 ThePoinc aréModelsofHyperbolicGeometry 149
ThePoincar´eUpperHalfPlaneModel ....................................................... 149
Vertical(Euclidean)Lines ............................................................................ 149
Isometries ........................................................................................................ 149
InversionintheCircle .................................................................................... 150
InversioninEuclideanGeometry ............................................................... 161
FractionalLinearTransformations ........................................................... 164
TheCrossRatio ................................................................................................ 169
Translations .................................................................................................... 173
Rotations .......................................................................................................... 177
Reflections ........................................................................................ 181
Lengths ............................................................................................................. 185
TheAxiomsofHyperbolicGeometry.......................................................... 186
TheAreaofTriangles...................................................................................... 186
The Poincar´eDisc Model ............................................................................ 188
CirclesandHorocycles .................................................................................. 190
HyperbolicTrigonometry............................................................................ 195
TheAngleofParallelism ................................................................................ 207
Curvature ......................................................................................................... 209
, 8 TilingsandLattices 211
RegularTilings ................................................................................................ 211
SemiregularTilings ....................................................................................... 211
LatticesandFundamentalDomains .......................................................... 212
TilingsinHyperbolicSpace .......................................................................... 212
And Modern Geometries With Computer
Activities 1st Edition By Arthur.
, A SURVEY OF CLASSICAL AND MODERN
GEOMETRIESWITH COMPUTER ACTIVITIES
1STEDITIONBYARTHURBARAGARSOLUTION
MANUAL
Contents
1 EuclideanGeometry 1
ThePythagoreanTheorem ............................................................................... 3
TheAxiomsofEuclideanGeometry ................................................................. 5
SSS,SAS,andASA .............................................................................................. 7
ParallelLines ..................................................................................................... 11
PonsAsinorum.................................................................................................. 12
TheStarTrekLemma ....................................................................................... 12
SimilarTriangles .............................................................................................. 18
PowerofthePoint ............................................................................................. 24
TheMediansandCentroid .............................................................................. 33
TheIncircle,Excircles,andtheLawofCosines............................................ 35
TheCircumcircleandLawofSines ................................................................ 42
TheEulerLine .................................................................................................... 48
TheNinePointCircle ........................................................................................ 50
PedalTrianglesandtheSimsonLine............................................................. 57
MenelausandCeva ........................................................................................... 67
2 GeometryinGreekAstronomy 75
TheRelativeSizeoftheMoonandSun ........................................................... 75
TheDiameteroftheEarth................................................................................ 76
3 ConstructionsUsingaCompassandStraightedge 81
TheRules ............................................................................................................ 81
SomeExamples ................................................................................................. 81
BasicResults ...................................................................................................... 82
TheAlgebraofConstructibleLengths.......................................................... 92
TheRegularPentagon ..................................................................................... 94
OtherConstructibleFigures ........................................................................ 102
, TrisectinganArbitraryAngle ...................................................................... 105
4 Geometer’sSketchpad 111
TheRulesofConstructions ........................................................................... 111
LemmasandTheorems................................................................................. 111
Archimedes’TrisectionAlgorithm ............................................................ 114
v
vi CONTENTS
VerificationofTheorems .............................................................................. 114
SophisticatedResults.................................................................................... 117
ParabolaPaper................................................................................................ 120
5 HigherDimensionalObjects 125
ThePlatonicSolids ......................................................................................... 125
TheDualityofPlatonicSolids .......................................................................127
TheEulerCharacteristic ............................................................................... 127
SemiregularPolyhedra................................................................................. 127
APartialCategorizationofSemiregularPolyhedra................................ 130
Four-DimensionalObjects .......................................................................... 138
6 HyperbolicGeometry 143
Models............................................................................................................... 143
ResultsfromNeutralGeometry .................................................................. 143
TheCongruenceofSimilarTriangles ......................................................... 145
ParallelandUltraparallelLines................................................................... 145
SinglyAsymptoticTriangles........................................................................ 146
DoublyandTriplyAsymptoticTriangles .................................................. 146
TheAreaofAsymptoticTriangles ............................................................... 147
7 ThePoinc aréModelsofHyperbolicGeometry 149
ThePoincar´eUpperHalfPlaneModel ....................................................... 149
Vertical(Euclidean)Lines ............................................................................ 149
Isometries ........................................................................................................ 149
InversionintheCircle .................................................................................... 150
InversioninEuclideanGeometry ............................................................... 161
FractionalLinearTransformations ........................................................... 164
TheCrossRatio ................................................................................................ 169
Translations .................................................................................................... 173
Rotations .......................................................................................................... 177
Reflections ........................................................................................ 181
Lengths ............................................................................................................. 185
TheAxiomsofHyperbolicGeometry.......................................................... 186
TheAreaofTriangles...................................................................................... 186
The Poincar´eDisc Model ............................................................................ 188
CirclesandHorocycles .................................................................................. 190
HyperbolicTrigonometry............................................................................ 195
TheAngleofParallelism ................................................................................ 207
Curvature ......................................................................................................... 209
, 8 TilingsandLattices 211
RegularTilings ................................................................................................ 211
SemiregularTilings ....................................................................................... 211
LatticesandFundamentalDomains .......................................................... 212
TilingsinHyperbolicSpace .......................................................................... 212
