This 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.
Author(s): Dan Pedoe
Edition: 2
Publisher: The Mathematical Association of America
Year: 1997
Language: English
Pages: 137
PREFACE......Page 7
CONTENTS......Page 9
CHAPTER 0......Page 11
1. The nine-point circle......Page 39
2. Inversion......Page 42
3. Feuerbach's theorem......Page 47
4. Extension of Ptolemy's theorem......Page 48
5. Fermat's problem......Page 49
6. The centres of similitude of two circles......Page 50
7. Coaxal systems of circles......Page 52
8. Canonical form for coaxal system......Page 54
9. Further properties......Page 57
10. Problem of Apollonius......Page 59
11. Compass geometry......Page 61
2. Euclidean three-space, E3......Page 64
3. First properties of the representation......Page 66
4. Coaxal systems......Page 67
5. Deductions from the representation......Page 68
6. Conjugacy relations......Page 70
7. Circles cutting at a given angle......Page 73
8. Representation of inversion......Page 74
9. The envelope of a system......Page 75
10. Some further applications......Page 77
11. Some anallagmatic curves......Page 81
2. The Argand diagram......Page 82
3. Modulus and argument......Page 83
4. Circles as level curves......Page 84
5. The cross-ratio of four complex numbers......Page 85
6. Mobius transformations of the a-plane......Page 88
7. A Mobius transformation dissected......Page 89
8. The group property......Page 91
10. The fundamental theorem......Page 93
11. The Poincar6 model......Page 96
13. Non-Euclidean distance......Page 99
1. Steiner's enlarging process......Page 102
2. Existence of a solution......Page 103
3. Method of solution......Page 104
4. Area of a polygon......Page 105
5. Regular polygons......Page 107
6. Rectifiable curves......Page 109
7. Approximation by polygons......Page 111
8. Area enclosed by a curve......Page 114
Exercises......Page 117
Solutions......Page 122
Appendix: Karl Wilhelm Feuerbach, Mathematician by Laura Guggenbuhl......Page 127
Index......Page 139
