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Excursions in Geometry


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Table of Contents

Introduction 1 A bit of background A practical problem A basic theorem Means 2 Harmonic division and Apollonian circles Harmonic conjugates The circle of Apollonius Coaxial families 3 Inversive geometry Transformations Inversion Invariants Cross-ratio 4 Application for inversive geometry Two easy problems Peaucellier's linkage Apollonius' problem Steiner chains The arbelos 5 The hexlet The conics defined A property of chains Soddy's hexlet Some new hexlets 6 The conic sections The reflection property Confocal conics Plan sections of a cone A characteristic of parabolas 7 Projective geometry Projective transformations The foundations Cross-ratio The complete quadrangle Pascal's Theorem Duality 8 Some Euclidean topics A navigation problem A three-circle problem The Euler line The nine-point circle A triangle problem 9 The golden section The pentagram Similarities and spirals The regular polyhedra The continued fraction for o 10 Angle trisection The unsolved problems of antiquity Other kinds of trisection 11 Some unsolved problems of modern geometry Convex sets and geometric inequalities Malfatti's problem The Kakeya problem Notes Index

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