Grand Valley State University

Circles, a mathematical view, Dan Pedoe

eng

Includes bibliographical references and index

Circles

33348979

Dan Pedoe

Spectrum series

a mathematical view

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

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 -- Representation of a circle -- Euclidean three-space, E₃ -- First properties of the representation -- Coaxal systems -- Deductions from the representation -- Conjugacy relations -- Circles cutting at a given angle -- Representation of inversion -- The envelope of a system -- Some further applications -- Some anallagmatic curves -- Complex numbers -- The Argand diagram -- Modulus and argument -- Circles as level curves -- The cross-ratio of four complex numbers -- Möbius tranformations of the s-plane -- A möbius transformation dissected -- The group property -- Special transformations -- The fundamental theorem -- The Poincaré model -- The parallel axiom -- Non-Euclidean distance -- 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