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| Preface | |
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| To the Student | |
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| Exploring Geometry | |
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| Overview | |
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| Discovery in Geometry | |
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| Variations on Two Familiar Geometric Themes | |
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| Discovery via the Computer | |
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| Steiner's Theorem | |
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| Foundations of Geometry 1: Points, Lines, Segments, Angles | |
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| Overview | |
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| An Introduction to Axiomatics and Proof | |
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| The Role of Examples and Models | |
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| Incidence Axioms for Geometry | |
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| Distance, Ruler Postulate, Segments, Rays, and Angles | |
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| Angle Measure and the Protractor Postulate | |
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| Plane Separation, Interior of Angles, Crossbar Theorem | |
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| Chapter Summary | |
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| Testing Your Knowledge | |
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| Foundations of Geometry 2: Triangles, Quadrilaterals, Circles | |
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| Overview | |
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| Triangles, Congruence Relations, SAS Hypothesis | |
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| Taxicab Geometry: Geometry without SAS Congruence | |
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| SAS, ASA, SSS Congruence, and Perpendicular Bisectors | |
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| Exterior Angle Inequality | |
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| The Inequality Theorems | |
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| Additional Congruence Criteria | |
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| Quadrilaterals | |
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| Circles | |
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| Chapter Summary | |
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| Testing Your Knowledge | |
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| Euclidean Geometry: Trigonometry, Coordinates and Vectors | |
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| Overview | |
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| Euclidean Parallelism, Existence of Rectangles | |
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| Parallelograms and Trapezoids: Parallel Projection | |
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| Similar Triangles, Pythagorean Theorem, Trigonometry | |
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| Regular Polygons and Tiling | |
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| The Circle Theorems | |
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| Euclid's Concept of Area and Volume | |
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| Coordinate Geometry and Vectors | |
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| Some Modern Geometry of the Triangle | |
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| Chapter Summary | |
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| Testing Your Knowledge | |
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| Transformations in Geometry | |
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| Overview | |
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| Euclid's Superposition Proof and Plane Transformations | |
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| Reflections: Building Blocks for Isometries | |
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| Translations, Rotations, and Other Isometries | |
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| Other Linear Transformations | |
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| Coordinate Characterizations | |
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| Transformation Groups | |
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| Using Tranformation Theory in Proofs | |
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| Chapter Summary | |
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| Testing Your Knowledge | |
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| Alternate Concepts for Parallelism: Non-Euclidean Geometry | |
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| Overview | |
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| Historical Background of Non-Euclidean Geometry | |
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| An Improbable Logical Case | |
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| Hyperbolic Geometry: Angle Sum Theorem | |
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| Two Models for Hyperbolic Geometry | |
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| Circular Inversion: Proof of SAS Postulate for Half-Plane Model | |
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| Chapter Summary | |
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| Testing Your Knowledge | |
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| An Introduction to Three-Dimensional Geometry | |
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| Overview | |
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| Orthogonality Concepts for Lines and Planes | |
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| Parallelism in Space: Prisms, Pyramids, and the Platonic Solids | |
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| Cones, Cylinders, and Spheres | |
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| Volume in E[superscript 3] | |
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| Coordinates, Vectors, and Isometries in E[superscript 3] | |
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| Spherical Geometry | |
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| Chapter Summary | |
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| Testing Your Knowledge | |
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| Appendixes | |
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| Bibliography | |
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| Review of Topics in Secondary School Geometry | |
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| The Geometer's Sketchpad: Brief Instructions | |
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| Unified Axiom System for the Three Classical Geometries | |
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| Answers to Selected Problems | |
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| Symbols, Definitions, Axioms, Theorems, and Corollaries | |
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| Index | |
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| Special Topics | |
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| An Introduction to Projective Geometry | |
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| An Introduction to Convexity Theory | |