Plane and Solid Geometry
This is a book on Euclidean geometry that covers the standard material in a completely new way, while also introducing a number of new topics that would be suitable as a junior-senior level undergraduate textbook. The author does not begin in the traditio
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J.M. Aarts
Plane and Solid Geometry
123
J.M. Aarts Delft University of Technology Mediamatics The Netherlands [email protected]
Translator: Reinie Ern´e Leiden, The Netherlands [email protected]
Editorial board: Sheldon Axler, San Francisco State University, San Francisco, CA, USA Vincenzo Capasso, University of Milan, Milan, Italy Carles Casacuberta, Universitat de Barcelona, Barcelona, Spain Angus MacIntyre, Queen Mary, University of London, London, UK Kenneth Ribet, University of California, Berkeley, CA, USA Claude Sabbah, Ecole Polytechnique, Palaiseau, France Endre Süli, Oxford University, Oxford, UK Wojbor Woyczynski, Case Western Reserve University, Cleveland, OH, USA
ISBN: 978-0-387-78240-9 e-ISBN: 978-0-387-78241-6 DOI: 10.1007/978-0-387-78241-6 Library of Congress Control Number: 2008935537 Mathematics Subject Classification (2000): 51-xx This is a translation of the Dutch, Meetkunde, originally published by Epsilon–Uitgaven, 2000. ¤ 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com
To Robert J. and Lucie P.
P L A N E G E O M E T R Y
1: 1.1–1.6
5 5.1
2: 2.1–2.6 5.2
3: 3.1–3.6
4: 4.1–4.6 5.4
5.5
5.3
5.6
S O L I D G E O M E T R Y
Preface
Nature and the world around us that we ourselves design, furnish, and build contain many geometric patterns and structures. This is one of the reasons that geometry should be studied at school. At first, the study of geometry is experimental. Results are taught and used in numerous examples. Only later do proofs come into play. But are these proofs truly necessary, or can we do without them? A natural answer is that every statement must be provided with a proof, because we want to know whether it is true. However, it is clear that the less experienced student may become frustrated by the presence of too many proofs. Only later will the student understand that proofs not only show the correctness of a statement, but also provide better insight into the relations among various properties of the objects that are being studied. Learning statements without proofs, you risk not being able to see the forest for the trees. For this reason, we will pay much attention to a careful presentation of proofs in this book. In the dev
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