Preface | p. ix |

To the Student | p. xvii |

Exploring Geometry | p. 1 |

Overview | p. 1 |

Discovery in Geometry | p. 2 |

Variations on Two Familiar Geometric Themes | p. 14 |

Discovery via the Computer | p. 27 |

Steiner's Theorem | p. 39 |

Foundations of Geometry 1: Points, Lines, Segments, Angles | p. 51 |

Overview | p. 51 |

An Introduction to Axiomatics and Proof | p. 52 |

The Role of Examples and Models | p. 62 |

Incidence Axioms for Geometry | p. 70 |

Distance, Ruler Postulate, Segments, Rays, and Angles | p. 77 |

Angle Measure and the Protractor Postulate | p. 90 |

Plane Separation, Interior of Angles, Crossbar Theorem | p. 103 |

Chapter Summary | p. 116 |

Testing Your Knowledge | p. 117 |

Foundations of Geometry 2: Triangles, Quadrilaterals, Circles | p. 119 |

Overview | p. 119 |

Triangles, Congruence Relations, SAS Hypothesis | p. 120 |

Taxicab Geometry: Geometry without SAS Congruence | p. 127 |

SAS, ASA, SSS Congruence, and Perpendicular Bisectors | p. 139 |

Exterior Angle Inequality | p. 152 |

The Inequality Theorems | p. 166 |

Additional Congruence Criteria | p. 174 |

Quadrilaterals | p. 183 |

Circles | p. 194 |

Chapter Summary | p. 208 |

Testing Your Knowledge | p. 209 |

Euclidean Geometry: Trigonometry, Coordinates and Vectors | p. 211 |

Overview | p. 211 |

Euclidean Parallelism, Existence of Rectangles | p. 211 |

Parallelograms and Trapezoids: Parallel Projection | p. 224 |

Similar Triangles, Pythagorean Theorem, Trigonometry | p. 236 |

Regular Polygons and Tiling | p. 254 |

The Circle Theorems | p. 269 |

Euclid's Concept of Area and Volume | p. 284 |

Coordinate Geometry and Vectors | p. 301 |

Some Modern Geometry of the Triangle | p. 315 |

Chapter Summary | p. 328 |

Testing Your Knowledge | p. 329 |

Transformations in Geometry | p. 331 |

Overview | p. 331 |

Euclid's Superposition Proof and Plane Transformations | p. 331 |

Reflections: Building Blocks for Isometries | p. 341 |

Translations, Rotations, and Other Isometries | p. 353 |

Other Linear Transformations | p. 362 |

Coordinate Characterizations | p. 373 |

Transformation Groups | p. 389 |

Using Tranformation Theory in Proofs | p. 402 |

Chapter Summary | p. 418 |

Testing Your Knowledge | p. 419 |

Alternate Concepts for Parallelism: Non-Euclidean Geometry | p. 421 |

Overview | p. 421 |

Historical Background of Non-Euclidean Geometry | p. 421 |

An Improbable Logical Case | p. 425 |

Hyperbolic Geometry: Angle Sum Theorem | p. 436 |

Two Models for Hyperbolic Geometry | p. 445 |

Circular Inversion: Proof of SAS Postulate for Half-Plane Model | p. 469 |

Chapter Summary | p. 489 |

Testing Your Knowledge | p. 490 |

An Introduction to Three-Dimensional Geometry | p. 493 |

Overview | p. 493 |

Orthogonality Concepts for Lines and Planes | p. 493 |

Parallelism in Space: Prisms, Pyramids, and the Platonic Solids | p. 503 |

Cones, Cylinders, and Spheres | p. 514 |

Volume in E[superscript 3] | p. 522 |

Coordinates, Vectors, and Isometries in E[superscript 3] | p. 532 |

Spherical Geometry | p. 545 |

Chapter Summary | p. 559 |

Testing Your Knowledge | p. 560 |

Appendixes | |

Bibliography | p. 1 |

Review of Topics in Secondary School Geometry | p. 2 |

The Geometer's Sketchpad: Brief Instructions | p. 27 |

Unified Axiom System for the Three Classical Geometries | p. 31 |

Answers to Selected Problems | p. 35 |

Symbols, Definitions, Axioms, Theorems, and Corollaries | p. 55 |

Index | p. 1 |

Special Topics | |

An Introduction to Projective Geometry | |

An Introduction to Convexity Theory | |

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