Preface xiii Our Motivation, Philosophy, and Pedagogy xiii Chapter Dependencies xv Supplements xvi Acknowledgments xvi To the Student xix ONE Using The Geometer's Sketchpad: Exploration and Conjecture 1 1.1 Discussion Part I: Getting Started with Sketchpad 2 1.2 Activities 3 1.3 Discussion Part II:Observation?Conjecture?Proof 5 Some Sketchpad Tips 6 Questions, Questions, Questions! 7 Language of Geometry 8 Euclid's Postulates 12 Congruence 14 Ideas About Betweenness 15 Constructions 16 Properties of Triangles 18 Properties of Quadrilaterals 19 Properties of Circles 20 Exploration and Conjecture: Inductive Reasoning 20 1.4 Exercises 21 1.5 Chapter Overview 24 TWO Mathematical Arguments and Triangle Geometry 29 2.1 Activities 30 2.2 Discussion 31 Deductive Reasoning 31 Rules of Logic 32 Conditional Statements: Implication 34 Mathematical Arguments 37 Universal and Existential Quantifiers 38 Negating a Quantified Statement 40 Congruence Criteria for Triangles 42 Concurrence Properties for Triangles 43 Brief Excursion into Circle Geometry 46 The Circumcircle of ABC 46 The Nine-Point Circle: A First Pass 47 Ceva's Theorem and Its Converse 47 Menelaus' Theorem and Its Converse 48 2.
3 Exercises 49 2.4 Chapter Overview 53 THREE Circle Geometry, Robust Constructions, and Proofs 57 3.1 Activities 58 3.2 Discussion 60 Axiom Systems: Ancient and Modern Approaches 60 Robust Constructions: Developing a Visual Proof 62 Step-by-Step Proofs 62 Incircles and Excircles 65 The Pythagorean Theorem 66 Language of Circles 67 Some Interesting Families of Circles 68 Power of a Point 70 Inversion in a Circle 71 The Arbelos and the Salinon 73 The Nine-Point Circle: A Second Pass 75 Methods of Proof 75 3.3 Exercises 78 3.4 Chapter Overview 82 FOUR Analytic Geometry 87 4.1 Activities 88 4.2 Discussion 90 Points 90 Lines 93 Distance 97 Using Coordinates in Proofs 100 Polar Coordinates 102 The Nine-Point Circle, Revisited 105 4.
3 Exercises 110 4.4 Chapter Overview 113 FIVE Taxicab Geometry 117 5.1 Activities 118 5.2 Discussion 122 An Axiom System for Metric Geometry 123 Circles 125 Ellipses 126 Measuring Distance from a Point to a Line 127 Parabolas 128 Hyperbolas 130 Axiom Systems 130 5.3 Exercises 131 5.4 Chapter Overview 132 SIX Transformational Geometry 135 6.1 Activities 136 6.2 Discussion 139 Transformations 139 Isometries 140 Composition of Isometries 144 Inverse Isometries 148 Using Isometries in Proofs 149 Isometries in Space 150 Inversion in a Circle, Revisited 151 6.
3 Exercises 155 6.4 Chapter Overview 158 SEVEN Isometries and Matrices 161 7.1 Activities 162 7.2 Discussion 164 Using Vectors to Represent Translations 164 Using Matrices to Represent Rotations 165 Using Matrices to Represent Reflections 166 Composition of Isometries 168 The General Form of a Matrix Representation 170 Using Matrices in Proofs 172 Similarity Transformations 174 7.3 Exercises 175 7.4 Chapter Overview 178 EIGHT Symmetry in the Plane 179 8.1 Activities 180 8.2 Discussion 183 Symmetries 183 Groups of Symmetries 184 Classifying Figures by Their Symmetries 186 Friezes and Symmetry 190 Wallpaper Symmetry 193 Tilings 194 8.
3 Exercises 198 8.4 Chapter Overview 200 NINE Hyperbolic Geometry 203 Part I: Exploring a New Universe 204 9.1 Activities 204 9.2 Discussion 207 Hyperbolic Lines and Segments 207 The Poincar´e Disk Model of the Hyperbolic Plane 207 Hype.