Circles: A Mathematical View

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Cambridge University Press, 1995 - Mathematics - 102 pages
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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. As a concession to the general neglect of geometry in school and college curricula, however, the author has supplemented this new edition with a chapter 0 designed to introduce readers to the special vocabulary of circle concepts with which the author could assume his readers of two generations ago were familiar. For example, Pedoe carefully explains what is meant by the circumcircle, incircle, and excircles of a triangle as well as the circumcentre, incentre, and otrthocentre. The reader can then understand his discussion in Chapter 1 of the nine-point circle, and of Feuerbach's theorem. As an appendix, Pedoe includes a biographical article by Laura Guggenbuhl on Karl Wilhelm Feuerbach, a little-known mathematician with a tragically short life, who published his theorem in a slender geometric treatise in 1822. Readers of Circles need only be armed with paper, pencil, compass and straightedge 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 time. And those who think that they learned all they needed to know about circles in high school 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 isoperimetric property of the circle. -- from back cover.
 

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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
Index
101
Copyright

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