PLAIN PLANE GEOMETRY PLAIN PLANE GEOMETRY Amol Sasane London School of Economics, UK Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Sasane, A. (Amol), 1976– Title: Plain plane geometry / Amol Sasane. Other titles: Plane geometry Description: New Jersey : World Scientific, 2016. | Includes bibliographical references and index. Identifiers: LCCN 2015039584 | ISBN 9789814740432 (hardcover : alk. paper) | ISBN 9789814740449 (softcover : alk. paper) Subjects: LCSH: Geometry, Plane--Study and teaching (Secondary) | Geometry, Plane--Study and teaching (Higher) | Geometry--Study and teaching (Secondary) | Geometry--Study and teaching (Higher) Classification: LCC QA455 .S24 2016 | DDC 516.22--dc23 LC record available at http://lccn.loc.gov/2015039584 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Cover design by Malin Christersson and Amol Sasane. A proof of Pythagoras’s Theorem by tessellation. Copyright © 2016 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any informa(cid:415)on storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore To Arun Preface Why this book? Because I want to share the feeling that planar geometry can be fun. Although as high school students, my genera(cid:415)on were taught Euclidean plane geometry, it is common to encounter a modern-day high school mathema(cid:415)cs textbook which is largely devoid of any pre(cid:425)y geometry theorems with proofs, as other important topics have replaced such old- fashioned things. The present book hopes to remedy this in some measure. What’s so special about geometry? We’ll elaborate this below, but essentially: (1) It showed that mathema(cid:415)cs involves not just numbers, but pictures (which is how many mathematicians think when creating new maths). (2) It convinced the student that mathema(cid:415)cs is beau(cid:415)ful and not boring. On the contrary, doing Mathematics can be enjoyable! (3) Besides all these lovely things, it taught students what a “proof” is, and prepared high school graduates for university level mathematics. The aim of this book is to cover the basics of the wonderful subject of Planar Geometry, at high school level, requiring no prerequisites beyond arithme(cid:415)c, and hopefully to convey the sense of joy which I had when I was taught geometry. While one might argue about the use of teaching such outdated things, and contrast this with teaching other useful things such as Cartesian geometry, algorithms and so on, surely, as men(cid:415)oned in item (3) above, there is one feature of Euclidean plane geometry which makes learning it worthwhile: it trains the student in understanding proofs, and also in devising one’s own proofs: in other words, it inculcates the very spirit of Mathematics! What is Planar Geometry? Planar figures are figures in the plane, such as triangles, quadrilaterals and circles. B y Planar Geometry, we mean a study of planar figures and their geometrical proper(cid:415)es. By a geometric property of a planar figure, we mean one which doesn’t change under a “rigid” mo(cid:415)on (that is mo(cid:415)on that does not distort the figure: examples of such mo(cid:415)on are rota(cid:415)on, transla(cid:415)on and reflec(cid:415)on). Examples of geometric proper(cid:415)es that don’t change are distances and angles. Why study Planar Geometry? Planar Geometry in some sense marks the birth of Mathema(cid:415)cs as it is prac(cid:415)ced in modern (cid:415)mes. O(cid:332)en it is considered as the first historical trea(cid:415)se (the works of Euclid) in which the rules of doing Mathema(cid:415)cs were set out in a systema(cid:415)c manner. If one asks: What is Mathema(cid:415)cs? Then the following is a rough answer: Mathema(cid:415)cs is a subject in which we make up defini(cid:415)ons and prove things about the defined objects in a logical manner. Thus Mathematics can be likened to a game, like Chess, in which there are: (1) objects (the chess board, chess pieces), (2) rules of the game (how the chess pieces move etc.), (3) the play (when two players actually play the game). In a similar manner in Mathematics, there are: (1) objects (definitions of mathematical objects), (2) rules (mathematical logic), (3) the play (making up theorems, that is, true statements about the objects and proving these truths). This endeavor of doing mathema(cid:415)cs is brought out very clearly for a school student by studying Planar Geometry à la Euclid, since (1) The objects in Planar Geometry are points, lines, angles, triangles, quadrilaterals and circles, and these are quite easy to understand since they are concrete, and visual. (2) The rules of the game are self-evident truths which are easy to accept, since they are again visual. These rules are also just a few in number and can be stated succinctly. Thus the logical structure of the proof becomes extremely clear to the student, when one gives reasons for the steps in the proof (such as usage of the Parallel Postulate, SAS Congruency Rule etc., which we will soon learn). (3) The theorems in Planar Geometry, and their proofs are par(cid:415)cularly beau(cid:415)ful and are ideal to convey the beauty of Mathema(cid:415)cs to the unini(cid:415)ated person. Indeed, prior to studying geometry, a school student’s exposure is mostly limited to arithmetic, mostly with no “soul” or sense of “play”. This changes dras(cid:415)cally with Planar Geometry, since everything is visual, so the mind’s eye can “see”, and also play (by drawing and trying this and that, guessing, experiencing an “aha!” moment, etc.). For example, in Exercise 3.27, we will encounter the following “proof without words”, of Pythagoras’s Theorem, saying that the square of the length of the biggest side in a right angled triangle is equal to the sum of the squares of the other two sides. What will we learn in this book? There are 5 chapters in the book: (1) Geometrical figures (2) Congruent triangles (3) Quadrilaterals (4) Similar triangles (5) Circles Perhaps the reader has encountered some or all of these objects before, and so might have some feeling of what is in store while reading this book. But besides this seemingly innocent backdrop of learning about planar geometry, the book has some ulterior motives: (1) to learn drawing pictures, and realizing that Mathema(cid:415)cs is not just about numbers, but can be