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 Morleys 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.
H. S. M. Coxeter, S. L. Greitzer
Preface
Chapter 1 Points and Lines Connected with a Triangle
1.1 The extended Law of Sines 1.2 Ceva´s theorem 1.3 Points of interest 1.4 The incircle and excircles 1.5 The Steiner-Lehmus theorem 1.6 The orthic triangle 1.7 The medial triangle and Euler line 1.8 The nine-point circle 1.9 Pedal triangles
Chapter 2 Some Properties of Circles 2.1 The power of a point with respect to a circle 2.2 The radical axis of two circles 2.3 Coaxal circles 2.4 More on the altitudes and orthocenter of a triangle 2.5 Simson lines 2.6 Ptolemy´s theorem and its extension 2.7 More on Simson lines 2.8 The Butterfly 2.9 Morley´s theorem
Chapter 3 Collinearity and Concurrence 3.1 Quadrangles; Varignon´s theorem 3.2 Cyclic quadrangles; Brahmagupta´s formula 3.3 Napoleon triangles 3.4 Menelaus´s theorem 3.5 Pappus´s theorem 3.6 Perspective triangles; Desargues´s theorem 3.7 Hexagons 3.8 Pascal´s theorem 3.9 Brianchon´s theorem
Chapter 4 Transformations 4.1 Translation 4.2 Rotation 4.3 Half-tum 4.4 Reflection 4.5 Fagnano´s problem 4.6 The three jug problem 4.7 Dilatation 4.8 Spiral similarity 4.9 A genealogy of transformations
Chapter 5 An Introduction to Inversive Geometry 5.1 Separation 5.2 Cross ratio 5.3 Inversion 5.4 The inversive plane 5.5 Orthogonality 5.6 Feuerbach´s theorem 5.7 Coaxal circles 5.8 Inversive distance 5.9 Hyperbolic functions
Chapter 6 An Introduction to Projective Geometry 6.1 Reciprocation 6.2 The polar circle of a triangle 6.3 Conics 138 6.4 Focus and directrix 6.5 The projective plane 6.6 Central conics 6.7 Stereographic and gnomonic projection Hints and Answers to Exercises References Glossary Index