9780065000061

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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College Geometry: A Discovery Approach

0065000064

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