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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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