The Algebra of Geometry

No part of this book may be reproduced or transmitted in any forms or by any means electrical or mechanical, including photocopying, recording, or any information storage or retrieval system without written permission from High-perception Ltd, except for the inclusion of quotations in a review, statutory exception, and to the provisions of collective licensing agreements. Introduction This is a book about points, lines, triangles and conics situated in a plane. The circle, being a special conic in the Euclidean plane, is given due prominence. In fact Chapter 6 is entirely devoted to the circle and to the properties of cyclic quadrilaterals. Before that I attempt to describe comprehensively the standard properties of triangles, quadrangles, hexagons, circles and con-ics. In the later chapters, the amount of material competing for possible inclusion is so immense that choices have had to be made. I decided to give no systematic account of triangle centres, which is a modern and competitive industry, now involving over three thousand points. I introduce a limited number of such centres, but only as they occur naturally in topics under discussion. A second choice is not to discuss in any detail the underlying groups of transformations that distinguish Euclidean, affine and projective geometry. These groups are of course mentioned, but as they have been the centre of attention in so many books during the twentieth century, it seems to me that one can be sure that readers are now well acquainted with the theoretical foundations of the subject, or, if not, that they have an abundance of splendid material to which they can refer. A third choice is to include nothing about cubic curves or curves of even higher degree. Some of these curves do, however, make an appearance in the examples and exercises. What I have chosen to include has been dictated to a large extent by two major considerations. These are, first and foremost, what the book is attempting to provide for readers. In short that is an account of the use of coordinates in Euclidean, affine and projective geometry, with the aim of enabling others to use such techniques efficiently. A second aim was to incorporate some of my main interests over the last fifteen years. Since I started to study geometry I have not become aware of any books written in English in recent years specialising in coordinate geometry. Sometimes one gets the impression that mathematicians.