(* Indicates optional section.)
Preface.
To the Student.
1. Exploring Geometry.
Discovery in Geometry.
Variations on Two Familiar Geometric Themes.
Discovery via the Computer.
Steiner's Theorem.
2. Foundations of Geometry I: Points, Lines, Segments, and Angles.
*An Introduction to Axiomatics and Proof.
*The Role of Examples and Models.
Incidence Axioms for Geometry.
Distance, Ruler Postulate, Segments, Rays, and Angles.
Angle Measure and the Protractor Postulate.
Plane Separation, Interior of Angles, Crossbar Theorem.
Chapter Summary.
3. Foundations of Geometry II: Triangles, Quadrilaterals, and Circles.
Triangles, Congruence Relations, SAS Hypothesis.
*Taxicab Geometry: Geometry Without SAS Congruence.
SAS, ASA, SSS Congruence, and Perpendicular Bisectors.
Exterior Angle Inequality.
The Inequality Theorems.
Additional Congruence Criteria.
Quadrilaterals.
Circles.
Chapter Summary.
4. Euclidean Geometry: Trigonometry, Coordinates, and Vectors.
Euclidean Parallelism, Existence of Rectangles.
Parallelograms and Trapezoids: Parallel Projection.
Similar Triangles, Pythagorean Theorem, Trigonometry.
Regular Polygons and Tiling.
*Euclid's Concept of Area and Volume.
Coordinate Geometry and Vectors.
*Some Modern Geometry of the Triangle.
Chapter Summary.
5. Transformations in Geometry.
Euclid's Superposition Proof and Plan Transformation.
Reflections: Building Blocks for Isometrics.
Translations, Rotations and Other Isometrics.
Other Linear Transformations.
Coordinate Characterization.
*Using Transformation Theory in Proofs.
Chapter Summary.
6. Alternative Concepts for Parallelism: Non-Euclidean Geometry.
Historical Background of Non-Euclidean Geometry.
An Improbable Logical Case.
Hyperbolic Geometry: Angle Sum Theorem.
*Two Models for Hyperbolic Geometry.
*Circular Inversion: Proof of SAS Postulate for Half-Plane Model.
Chapter Summary.
7. An Introduction to Three Dimensional Geometry.
Orthogonality Concepts for Lines and Plans.
Parallelism in Space: Prisms, Pyramids, and the Platonic Solids.
Cones, Cylinders, and Spheres.
Volume in E3.
Coordinates, Vectors, and Isometries in E3.
Spherical Geometry.
Chapter Summary.
Appendixes.
A. Bibliography.
B. Review of Topics in Secondary School Geometry.
C. The Geometer's Sketchpad: Brief Instructions.
D. Unified Axiom System for the Three Classical Geometries.
E. Answers to Selected Problems.
F. Symbols, Definitions, Axioms, Theorems, and Corollaries.
Special Topics: An Introduction to Projective Geometry.
An Introduction to Convexity Theory.
