Preface |
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ix | |
To the Student |
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xvii | |
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1 | (50) |
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1 | (1) |
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2 | (12) |
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Variations on Two Familiar Geometric Themes |
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14 | (13) |
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Discovery via the Computer |
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27 | (12) |
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39 | (12) |
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Foundations of Geometry 1: Points, Lines, Segments, Angles |
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51 | (68) |
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51 | (1) |
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An Introduction to Axiomatics and Proof |
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52 | (10) |
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The Role of Examples and Models |
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62 | (8) |
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Incidence Axioms for Geometry |
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70 | (7) |
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Distance, Ruler Postulate, Segments, Rays, and Angles |
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77 | (13) |
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Angle Measure and the Protractor Postulate |
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90 | (13) |
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Plane Separation, Interior of Angles, Crossbar Theorem |
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103 | (16) |
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116 | (1) |
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117 | (2) |
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Foundations of Geometry 2: Triangles, Quadrilaterals, Circles |
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119 | (92) |
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119 | (1) |
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Triangles, Congruence Relations, SAS Hypothesis |
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120 | (7) |
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Taxicab Geometry: Geometry without SAS Congruence |
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127 | (12) |
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SAS, ASA, SSS Congruence, and Perpendicular Bisectors |
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139 | (13) |
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Exterior Angle Inequality |
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152 | (14) |
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166 | (8) |
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Additional Congruence Criteria |
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174 | (9) |
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183 | (11) |
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194 | (17) |
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208 | (1) |
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209 | (2) |
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Euclidean Geometry: Trigonometry, Coordinates and Vectors |
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211 | (120) |
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211 | (1) |
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Euclidean Parallelism, Existence of Rectangles |
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211 | (13) |
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Parallelograms and Trapezoids: Parallel Projection |
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224 | (12) |
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Similar Triangles, Pythagorean Theorem, Trigonometry |
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236 | (18) |
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Regular Polygons and Tiling |
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254 | (15) |
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269 | (15) |
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Euclid's Concept of Area and Volume |
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284 | (17) |
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Coordinate Geometry and Vectors |
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301 | (14) |
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Some Modern Geometry of the Triangle |
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315 | (16) |
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328 | (1) |
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329 | (2) |
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Transformations in Geometry |
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331 | (90) |
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331 | (1) |
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Euclid's Superposition Proof and Plane Transformations |
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331 | (10) |
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Reflections: Building Blocks for Isometries |
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341 | (12) |
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Translations, Rotations, and Other Isometries |
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353 | (9) |
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Other Linear Transformations |
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362 | (11) |
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Coordinate Characterizations |
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373 | (16) |
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389 | (13) |
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Using Transformation Theory in Proofs |
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402 | (19) |
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418 | (1) |
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419 | (2) |
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Alternate Concepts for Parallelism: Non-Euclidean Geometry |
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421 | (72) |
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421 | (1) |
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Historical Background of Non-Euclidean Geometry |
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421 | (4) |
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An Improbable Logical Case |
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425 | (11) |
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Hyperbolic Geometry: Angle Sum Theorem |
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436 | (9) |
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Two Models for Hyperbolic Geometry |
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445 | (24) |
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Circular Inversion: Proof of SAS Postulate for Half-Plane Model |
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469 | (24) |
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489 | (1) |
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490 | (3) |
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An Introduction to Three-Dimensional Geometry |
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493 | (67) |
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493 | (1) |
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Orthogonality Concepts for Lines and Planes |
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493 | (10) |
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Parallelism in Space: Prisms, Pyramids, and the Platonic Solids |
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503 | (11) |
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Cones, Cylinders, and Spheres |
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514 | (8) |
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522 | (10) |
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Coordinates, Vectors, and Isometries in E3 |
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532 | (13) |
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545 | (15) |
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559 | (1) |
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560 | |
Appendixes |
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A-1 | |
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Appendix B Review of Topics in Secondary School Geometry |
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A-2 | |
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Appendix C The Geometer's Sketchpad: Brief Instructions |
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A-27 | |
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Appendix D Unified Axiom System for the Three Classical Geometries |
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A-31 | |
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Appendix E Answers to Selected Problems |
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A-35 | |
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Appendix F Symbols, Definitions, Axioms, Theorems, and Corollaries |
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A-55 | |
Index |
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I-1 | |