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Among the many beautiful and nontrivial theorems in geometry found in Geometry Revisited are the theorems of Ceva, Menelaus, Pappus, Desargues, Pascal, and Brianchon. A nice proof is given of Morley's remarkable theorem on angle trisectors. The transformational point of view is emphasized: reflections, rotations, translations, similarities, inversions, and affine and projective transformations. Many fascinating properties of circles, triangles, quadrilaterals, and conics are developed. Front Cover 1 Geometry Revisited 4 Copyright Page 5 Contents 14 Preface 12 Chapter 1. Points and Lines Connected with a Triangle 16 1.1 The extended Law of Sines 16 1.2 Ceva’s theorem 19 1.3 Points of interest 22 1.4 The incircle and excircles 26 1.5 The Steiner-Lehmus theorem 29 1.6 The orthic triangle 31 1.7 The medial triangle and Euler line 33 1.8 The nine-point Circle 35 1.9 Pedal triangles 37 Chapter 2. Some Properties of Circles 42 2.1 The power of a point with respect to a circle 42 2.2 The radical axis of two circles 46 2.3 Coaxal circles 50 2.4 More on the altitudes and orthocenter of a triangle 51 2.5 Simson lines 55 2.6 Ptolemy’s theorem and its extension 57 2.7 More on Simson lines 58 2.8 The Butterfly 60 2.9 Morley’s theorem 62 Chapter 3. Collinearity and Concurrence 66 3.1 Quadrangles; Varignon’s theorem 66 3.2 Cyclic quadrangles; Brahmagupta’s formula 71 3.3 Napoleon triangles 75 3.4 Menelaus’s theorem 81 3.5 Pappus’s theorem 82 3.6 Perspective triangles; Desargues’s theorem 85 3.7 Hexagons 88 3.8 Pascal’s theorem 89 3.9 Brianchon’s theorem 92 Chapter 4. Transformations 95 4.1 Translation 96 4.2 Rotation 97 4.3 Half-turn 100 4.4 Reflection 101 4.5 Fagnano’s problem 103 4.6 The three jug problem 104 4.7 Dilatation 109 4.8 Spiral similarity 110 4.9 A genealogy of transformations 115 Chapter 5. An Introduction to Inversive Geometry 118 5.1 Separation 118 5.2 Cross ratio 122 5.3 Inversion 123 5.4 The inversive plane 127 5.5 Orthogonality 129 5.6 Feuerbach’s theorem 132 5.7 Coaxal circles 135 5.8 Inversive distance 138 5.9 Hyperbolic functions 141 Chapter 6. An Introduction to Projective Geometry 147 6.1 Reciprocation 147 6.2 The polar circle of a triangle 151 6.3 Conics 153 6.4 Focus and directrix 156 6.5 The projective plane 159 6.6 Central conics 161 6.7 Stereographic and gnomonic projection 165 Hints and Answers to Exercises 169 References 196 Glossary 198 Index 204 Back Cover 210 Annotation Among the many beautiful and nontrivial theorems in geometry found in Geometry Revisited are the theorems of Ceva, Menelaus, Pappus, Desargues, Pascal, and Brianchon. A nice proof is given of Morley's remarkable theorem on angle trisectors. The transformational point of view is emphasized: reflections, rotations, translations, similarities, inversions, and affine and projective transformations. Many fascinating properties of circles, triangles, quadrilaterals, and conics are developed Among the many beautiful and nontrivial theorems in geometry found here are the theorems of Ceva, Menelaus, Pappus, Desargues, Pascal, and Brianchon. A nice proof is given of Morley's remarkable theorem on angle trisectors. The transformational point of view is emphasized: reflections, rotations, translations, similarities, inversions, and affine and projective transformations. Many fascinating properties of circles, triangles, quadrilaterals, and conics are developed The purpose of this chapter is to recall some of these half-forgotten things to which Dr. Bell referred, to derive some new theorems, developed since Euclid, and to apply our findings to interesting situation.