Roads to geometry

1. Rules of the Road (Axiomatic Systems). Historical Background: Axiomatic Systems and their Properties. Finite Geometries. Axioms for Incidence Geometry. 2. Many Ways to Go. Introduction. Euclid's Geometry and Euclid's Elements. An Introduction to Modern Euclidean Geometries. Hilbert's Model for Euclidean Geometry. Birkhoff's Model for Euclidean Geometry. SMSG Postulates for Euclidean Geometry. Non-Euclidean Geometries. 3. Traveling Together (Neutral Geometry). Introduction. Preliminary Notions. Congruence Conditions. The Place of Parallels. The Saccheri-Legendre Theorem. The Search for a Rectangle. Summary. 4. One Way to Go (Euclidean Geometry of the Plane). Introduction. The Parallel Postulate and Some Implications. Congruence and Area. Similarity. Euclidean Results Concerning Circles. Some Euclidean Results Concerning Triangles. More Euclidean Results Concerning Triangles. The Nine-Point Circle. Euclidean Constructions. Summary. 5. Side Trips (Analytic and Transformational Geometry). Introduction. Analytic Geometry. Transformational Geometry. Analytic Transformations. Inversion. Summary. 6. Other Ways to Go (Non-Euclidean Geometries). Introduction. A Return to Neutral Geometry: The Angle of Parallelism. The Hyperbolic Parallel Postulate. Hyperbolic Results Concerning Polygons. Area in Hyperbolic Geometry. Showing Consistency: A Model for Hyperbolic Geometry. Classifying Theorems. Elliptic Geometry: A Geometry With No Parallels? Geometry in the Real World. Summary. 7. All Roads Lead To ... Projective Geometry. Introduction. The Real Projective Plane. Duality. Perspectivity. The Theorem of Desargues. Projective Transformations. Summary. Appendix A. Euclid's Definitions and Postulates Book I. Appendix B. Hilbert's Axioms for Euclidean Plane Geometry. Appendix C. Birkhoff's Postulates for Euclidean Plane Geometry. Appendix D. The SMSG Postulates for Euclidean Geometry. Appendix E. Geometer's SketchPad... Scripts for Poincare Model of Hyperbolic Geometry. Bibliography. Index.