Circles

Table of Contents

• Preface
• 1. The nine-point circle, inversion, Feuerbach's theorem, extension of Ptolemy's theorem, Fermat's problem, the centres of similitude of two circles, coaxal systems of circles, canonical form for coaxal system, further properties, problem of Apollonius, compass geometry
• 2. Representation of a circle, Euclidean three-space, first properties of the representation, coaxal systems, deductions from the representation 0, conjugacy relations, circles cutting at a given angle, representation of inversion, the envelope of a system, some further applications, some anallagmatic curves
• 3. Complex numbers, the Argand diagram, modulus and argument, circles as level curves, the cross-ratio of four complex numbers, Moebius transformations of the s-plane, a Moebius transformation dissected, the group property, special transformations, the fundamental theorem, the Poincare model, the parallel axiom, non-Euclidean distance
• 4. Steiner's enlarging process, existence of a solution, method of solution, area of a polygon, regular polygons, rectifiable curves, approximation by polygons, area enclosed by a curve
• Exercises
• Solutions
• Appendix: Karl Wilhelm Feuerbach, Mathematician, by Laura Guggenbuhl.
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