## Circles: A Mathematical ViewThis revised edition of a mathematical classic originally published in 1957 will bring to a new generation of students the enjoyment of investigating that simplest of mathematical figures, the circle. The author has supplemented this new edition with a special chapter designed to introduce readers to the vocabulary of circle concepts with which the readers of two generations ago were familiar. Readers of Circles need only be armed with paper, pencil, compass, and straight edge to find great pleasure in following the constructions and theorems. Those who think that geometry using Euclidean tools died out with the ancient Greeks will be pleasantly surprised to learn many interesting results which were only discovered in modern times. Novices and experts alike will find much to enlighten them in chapters dealing with the representation of a circle by a point in three-space, a model for non-Euclidean geometry, and the isoperimetric property of the circle. |

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### Contents

CHAPTER I | 1 |

Inversion | 4 |

Feuerbachs theorem | 9 |

Extension of Ptolemys theorem | 10 |

Fermats problem | 11 |

The centres of similitude of two circles | 12 |

Coaxal systems of circles | 14 |

Canonical form for coaxal system | 16 |

Modulus and argument | 45 |

Circles as level curves | 46 |

The crossratio of four complex numbers | 47 |

Mobius transformations of the flplane | 50 |

A Mobius transformation dissected | 51 |

The group property | 53 |

Special transformations | 55 |

The Poincar6 model | 58 |

Further properties | 19 |

Problem of Apollonius | 22 |

Compass geometry | 23 |

Representation of a circle | 26 |

First properties of the representation | 28 |

Coaxal systems | 29 |

Deductions from the representation | 30 |

Conjugacy relations | 33 |

Circles cutting at a given angle | 35 |

Representation of inversion | 36 |

The envelope of a system | 37 |

Some further applications | 39 |

Some anallagmatic curves | 43 |

CHAPTER III | 44 |

The parallel axiom | 61 |

CHAPTER IV | 64 |

Existence of a solution | 65 |

Method of solution | 66 |

Area of a polygon | 67 |

Regular polygons | 69 |

Rectifiable curves | 71 |

Approximation by polygons | 73 |

Area enclosed by a curve | 76 |

Exercises | 79 |

Solutions | 84 |

Karl Wilhelm Feuerbach Mathematician by Laura Guggenbuhl | 89 |

101 | |

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### Common terms and phrases

7i-line A-line bisector bisects called centre of inversion centre of similitude Ceva theorem chapter circle centre circle orthogonal circle touches circle which touches circle-tangential equation circumcircle closed curve co-arc coaxal system coaxal system determined compass complex numbers conic system construction cross-ratio D[ab draw equal equilateral polygon Erlangen Euclidean geometry excircles Figure fixed point given point Hence inscribed intersecting type Karl Wilhelm Feuerbach length limiting points line of centres locus M-transformation which maps Mathematical meets Q Menelaus theorem midpoint Mobius transformation modulus nine-point circle non-Euclidean non-intersecting obtain pairs parabola parallel axiom parallel lines passes perimeter perpendicular plane Oxy point of contact point of intersection point-circle polar plane problem of Apollonius projection proof prove quartic curve radical axis radii ratio represented respect right angles segment sides straight line system of circles tangent theorem three circles three given circles triangle ABC vertex zero