College Geometry

Preface. Our Motivation, Philosophy, and Pedagogy. Chapter Dependencies. Supplements. Acknowledgments. To the Student. ONE Using The Geometer's Sketchpad: Exploration and Conjecture. 1.1 Discussion Part I: Getting Started with Sketchpad. 1.2 Activities. 1.3 Discussion. Part II: Observation - Conjecture - Proof. Some Sketchpad Tips. Questions, Questions, Questions! Language of Geometry. Euclid's Postulates. Congruence. Ideas About Betweenness. Constructions. Properties of Triangles. Properties of Quadrilaterals. Properties of Circles. Exploration and Conjecture: Inductive Reasoning. 1.4 Exercises. 1.5 Chapter Overview. TWO Mathematical Arguments and Triangle Geometry. 2.1 Activities. 2.2 Discussion. Deductive Reasoning. Rules of Logic. Conditional Statements: Implication. Mathematical Arguments. Universal and Existential Quantifiers. Negating a Quantified Statement. Congruence Criteria for Triangles. Concurrence Properties for Triangles. Brief Excursion into Circle Geometry. The Circumcircle of ABC. The Nine-Point Circle: A First Pass. Ceva's Theorem and Its Converse. Menelaus' Theorem and Its Converse. 2.3 Exercises. 2.4 Chapter Overview. THREE Circle Geometry, Robust Constructions, and Proofs. 3.1 Activities. 3.2 Discussion. Axiom Systems: Ancient and Modern Approaches. Robust Constructions: Developing a Visual Proof. Step-by-Step Proofs. Incircles and Excircles. The Pythagorean Theorem. Language of Circles. Some Interesting Families of Circles. Power of a Point. Inversion in a Circle. The Arbelos and the Salinon. The Nine-Point Circle: A Second Pass. Methods of Proof. 3.3 Exercises. 3.4 Chapter Overview. FOUR Analytic Geometry. 4.1 Activities. 4.2 Discussion. Points. Lines. Distance. Using Coordinates in Proofs. Polar Coordinates. The Nine-Point Circle, Revisited. 4.3 Exercises. 4.4 Chapter Overview. FIVE Taxicab Geometry. 5.1 Activities. 5.2 Discussion. An Axiom System for Metric Geometry. Circles. Ellipses. Measuring Distance from a Point to a Line. Parabolas. Hyperbolas. Axiom Systems. 5.3 Exercises. 5.4 Chapter Overview. SIX Transformational Geometry. 6.1 Activities. 6.2 Discussion. Transformations. Isometries. Composition of Isometries. Inverse Isometries. Using Isometries in Proofs. Isometries in Space. Inversion in a Circle, Revisited. 6.3 Exercises. 6.4 Chapter Overview. SEVEN Isometries and Matrices. 7.1 Activities. 7.2 Discussion. Using Vectors to Represent Translations. Using Matrices to Represent Rotations. Using Matrices to Represent Reflections. Composition of Isometries. The General Form of a Matrix Representation. Using Matrices in Proofs. Similarity Transformations. 7.3 Exercises. 7.4 Chapter Overview. EIGHT Symmetry in the Plane. 8.1 Activities. 8.2 Discussion. Symmetries. Groups of Symmetries. Classifying Figures by Their Symmetries. Friezes and Symmetry. Wallpaper Symmetry. Tilings. 8.3 Exercises. 8.4 Chapter Overview. NINE Hyperbolic Geometry. Part I: Exploring a New Universe. 9.1 Activities. 9.2 Discussion. Hyperbolic Lines and Segments. The Poincare Disk Model of the Hyperbolic Plane. Hyperbolic Triangles. Hyperbolic Circles. Measuring Distance in the Poincare Disk Model. Circumcircles and Incircles of Hyperbolic Triangles. Congruence of Triangles in the Hyperbolic Plane. Part II: The Parallel Postulate in the Poincar-e Disk. 9.3 Activities. 9.4 Discussion. The Hyperbolic and Elliptic Parallel Postulates. Parallel Lines in the Hyperbolic Plane. Quadrilaterals in the Hyperbolic Plane. 9.5 Exercises. 9.6 Chapter Overview. TEN Projective Geometry. 10.1 Activities. 10.2 Discussion. An Axiom System. Models for the Projective Plane. Duality. Coordinates for Projective Geometry. Projective Transformations. 10.3 Exercises. 10.4 Chapter Overview. Appendix A Trigonometry. A.1 Activities. A.2 Discussion. Right Triangle Trigonometry. Unit Circle Trigonometry. Solving Trigonometric Equations. Double Angle Formulas. Angle Sum Formulas. Half-Angle Formulas. The Law of Sines and the Law of Cosines. A.3 Exercises. Appendix B Calculating with Matrices. B.1 Activities. B.2 Discussion. Linear Combinations of Vectors. Dot Product of Vectors. Multiplying a Matrix by a Vector. Multiplying Two Matrices. The Determinant of a Matrix. B.3 Exercises. Bibliography. Index.