Preface |
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xiii | |
Especially for Students |
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xiii | |
Notes for Instructors |
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xv | |
Our Motivation, Philosophy, and Pedagogy |
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xv | |
Prerequisites and Chapter Dependencies |
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xviii | |
Acknowledgments |
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xix | |
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1 | (21) |
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1.1 Activities: Getting Started with GeoGebra |
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2 | (7) |
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1.2 Discussion: Exploring and Conjecturing |
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9 | (8) |
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9 | (1) |
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Constructing → Exploring → Conjecturing: Inductive Reasoning |
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9 | (1) |
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10 | (1) |
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Explorations, Observations, Questions |
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11 | (2) |
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The Family of Quadrilaterals |
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13 | (2) |
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Angles Inscribed in Circles |
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15 | (1) |
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16 | (1) |
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17 | (3) |
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20 | (2) |
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Two Constructing → Proving |
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22 | (31) |
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23 | (3) |
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2.2 Discussion: Euclid's Postulates and Constructions |
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26 | (18) |
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26 | (1) |
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Congruence and Similarity |
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27 | (3) |
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30 | (2) |
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Geometric Language Revisited |
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32 | (1) |
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Conditional Statements: Implication |
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33 | (2) |
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Using Robust Constructions to Develop a Proof |
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35 | (1) |
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Angles and Measuring Angles |
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36 | (2) |
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Constructing Perpendicular and Parallel Lines |
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38 | (1) |
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38 | (2) |
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Euclid's Parallel Postulate |
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40 | (1) |
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Euclid's Constructions in the Elements |
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41 | (2) |
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43 | (1) |
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44 | (4) |
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48 | (5) |
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Three Mathematical Arguments and Triangle Geometry |
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53 | (31) |
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54 | (2) |
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56 | (19) |
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56 | (2) |
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Universal and Existential Quantifiers |
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58 | (1) |
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Negating a Quantified Statement |
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59 | (1) |
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Direct Proof and Disproof by Counterexample |
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60 | (2) |
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62 | (3) |
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Congruence Criteria for Triangles |
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65 | (2) |
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The Converse and the Contrapositive |
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67 | (2) |
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Concurrence Properties for Triangles |
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69 | (2) |
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Ceva's Theorem and its Converse |
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71 | (2) |
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Brief Excursion into Circle Geometry |
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73 | (1) |
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74 | (1) |
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The Nine-Point Circle: A First Pass |
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74 | (1) |
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Menelaus' Theorem and Its Converse |
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75 | (1) |
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75 | (6) |
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81 | (3) |
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Four Circle Geometry and Proofs |
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84 | (26) |
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85 | (2) |
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87 | (16) |
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Axiom Systems: Ancient and Modern Approaches |
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87 | (2) |
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89 | (1) |
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89 | (1) |
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90 | (1) |
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Additional Methods of Proof |
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91 | (2) |
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93 | (2) |
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95 | (1) |
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Some Interesting Families of Circles |
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96 | (1) |
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The Arbelos and the Salinon |
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97 | (2) |
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99 | (1) |
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100 | (2) |
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The Nine-Point Circle: A Second Pass |
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102 | (1) |
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103 | (4) |
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107 | (3) |
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110 | (31) |
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111 | (3) |
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114 | (22) |
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114 | (3) |
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117 | (5) |
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122 | (3) |
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Using Coordinates in Proofs |
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125 | (2) |
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Another Look at the Radical Axis |
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127 | (1) |
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128 | (3) |
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The Nine-Point Circle, Revisited |
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131 | (5) |
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136 | (3) |
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139 | (2) |
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141 | (19) |
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142 | (5) |
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147 | (8) |
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An Axiom System for Metric Geometry |
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147 | (3) |
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150 | (1) |
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151 | (1) |
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Measuring Distance from a Point to a Line |
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152 | (1) |
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153 | (1) |
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154 | (1) |
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155 | (1) |
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155 | (3) |
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158 | (2) |
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160 | (23) |
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161 | (2) |
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163 | (17) |
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An Axiom System for an Affine Plane |
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163 | (4) |
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An Axiom System for a Projective Plane |
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167 | (3) |
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170 | (3) |
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Relating Affine Planes to Projective Planes |
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173 | (3) |
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Coordinates for Finite Geometries |
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176 | (4) |
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180 | (2) |
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182 | (1) |
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Eight Transformational Geometry |
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183 | (23) |
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184 | (2) |
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186 | (14) |
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187 | (1) |
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187 | (5) |
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192 | (1) |
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Composition of Isometries |
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193 | (3) |
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196 | (2) |
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Using Isometries in Proofs |
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198 | (1) |
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199 | (1) |
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200 | (3) |
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203 | (3) |
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Nine Isometries and Matrices |
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206 | (18) |
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207 | (2) |
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209 | (11) |
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Using Vectors to Represent Translations |
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209 | (1) |
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Using Matrices to Represent Rotations |
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210 | (1) |
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Using Matrices to Represent Reflections |
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211 | (2) |
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Composition of Isometries |
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213 | (2) |
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The General Form of a Matrix Representation |
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215 | (2) |
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217 | (2) |
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Similarity Transformations |
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219 | (1) |
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220 | (3) |
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223 | (1) |
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Ten Symmetry in the Plane |
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224 | (24) |
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225 | (3) |
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228 | (16) |
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229 | (1) |
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230 | (2) |
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Classifying Figures by Their Symmetries |
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232 | (4) |
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236 | (3) |
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239 | (1) |
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239 | (5) |
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244 | (2) |
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246 | (2) |
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Eleven Hyperbolic Geometry |
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248 | (34) |
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Part I Exploring a New Universe |
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250 | (1) |
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250 | (3) |
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253 | (10) |
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Hyperbolic Lines and Segments |
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253 | (1) |
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The Poincare Disk Model of the Hyperbolic Plane |
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254 | (2) |
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Measuring Distance in the Poincare Disk Model |
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256 | (2) |
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258 | (1) |
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258 | (2) |
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Circumcircles and Incircles of Hyperbolic Triangles |
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260 | (1) |
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Congruence of Triangles in the Hyperbolic Plane |
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261 | (2) |
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Part II The Parallel Postulate in Hyperbolic Geometry |
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263 | (1) |
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263 | (2) |
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265 | (10) |
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The Hyperbolic and Elliptic Parallel Postulates |
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265 | (2) |
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267 | (2) |
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The Exterior Angle Theorem |
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269 | (2) |
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Quadrilaterals in the Hyperbolic Plane |
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271 | (2) |
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Another Look at Triangles in the Hyperbolic Plane |
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273 | (1) |
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Area in the Hyperbolic Plane |
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274 | (1) |
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275 | (4) |
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279 | (3) |
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Twelve Projective Geometry |
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282 | (1) |
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283 | (2) |
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285 | (1) |
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285 | (1) |
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Models for the Projective Plane |
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286 | (5) |
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291 | (5) |
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Coordinates for Projective Geometry |
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296 | (6) |
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Projective Transformations |
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302 | (5) |
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307 | (3) |
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310 | |
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1 | (1) |
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1 | (3) |
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4 | (7) |
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Right Triangle Trigonometry |
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4 | (1) |
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5 | (2) |
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Solving Trigonometric Equations |
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7 | (1) |
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7 | (1) |
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8 | (2) |
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10 | (1) |
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The Law of Sines and the Law of Cosines |
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10 | (1) |
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11 | (2) |
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APPENDIX B Calculating with Matrices |
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13 | (1) |
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13 | (2) |
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15 | (4) |
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Linear Combinations of Vectors |
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15 | (1) |
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16 | (1) |
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Multiplying a Matrix Times a Vector |
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16 | (1) |
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17 | (1) |
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The Determinant of a Matrix |
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18 | (1) |
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19 | |
Bibliography |
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1 | (1) |
Index |
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1 | |