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A Basic Assumptions In this chapter, we repeat the basic assumptions for plane and solid geometry, plus a number of definitions. Definition. A metric space is a set M with a metric d. We call the elements of M the points of the metric space. To every pair of points X and Y ,the metric d associates a real number, the distance from X to Y . This distance satisfies the following properties: for every X, Y , Z in M, 1. d(X, Y ) ≥ 0; 2. d(X, Y ) = 0 if and only if X = Y ; 3. d(X, Y )=d(Y,X); 4. d(X, Z) ≤ d(X, Y )+d(Y,Z). Basic Assumption 1 The Euclidean plane V and 3-space W are both metric spaces containing more than one point. Definition. For any pair of points A and B in a metric space M with metric d, the line segment [AB]isgivenby [AB]={ X ∈ M : d(A, X)+d(X, B)=d(A, B) } . If A and B are distinct, the line AB is given by AB = { X : X ∈ [AB]orB ∈ [AX]orA ∈ [XB] } . Basic Assumption 2 Every line l in the plane or in 3-space admits an iso- metric surjection ϕ from l to R. Basic Assumption 3 The Euclidean plane V and 3-space W each contain three noncollinear points. 334 A Basic Assumptions Definition. For lines in the plane V we define the notions parallel and inter- secting. We say that lines m and n are parallel if m = n or m ∩ n = ∅.Ifm and n are not parallel, we say that m and n are intersecting lines. We call the point P the intersection point of the lines l and m if P lies on both l and m. Basic Assumption 4 For every point P and every line m of the plane V , there exists a unique line passing through P and parallel to m. Definition. Let l and m be intersecting lines of the plane V with common point C.Wesaythatl is perpendicular to m if for every point A of l and every point B of m we have d(C, A)2 + d(C, B)2 = d(A, B)2 . Basic Assumption 5 For every point P and every line m of the plane V , there exists a line l through P that is perpendicular to m. In the plane V , two intersecting lines l and m define four angles.IfP is the intersection point of the lines, any two points A and B other than P on l and m, respectively, determine exactly one of these angles. Every angle has an interior. Definition. We say that an angle P is congruent to an angle Q if there is an isometric surjection from angle P onto angle Q. Basic Assumption 6 The intersection of any angle P in the plane V and the circle (P, r) with center P and radius r is an arc. If r =1, we denote the length of this arc by ∠P ;ifr = R, the length of the arc is equal to R × ∠P . Moreover, we have the following properties: 1. 0 < ∠P<πfor every angle P . 2. If angles P and Q are congruent, then ∠P = ∠Q. 3. For every real number c satisfying 0 <c<∠AP B, there exists a point Q in the interior of angle AP B such that c = ∠AP Q. 4. If a point Q lies in the interior of angle AP B,then ∠AP B = ∠AP Q + ∠QP B . Definition. Let A, B,andC be three noncollinear points in W .Theplane α through A, B,andC is the subset of W with the following properties: 1. A, B,andC lie in α. 2. If D and E are distinct points of α, then the line DE lies completely inside α. 3. A point X lies in α if and only if there is a series of points X1, X2,...,Xn = X such that for 1 ≤ i ≤ n the point Xi is either equal to A, B,orC, or lies on the line XkXl for some 1 ≤ k<i,1≤ l<i. A Basic Assumptions 335 Basic Assumption 7 For every plane α in W there is an isometric surjec- tion ϕ from α onto the Euclidean plane V . Definition. Two noncoplanar lines in W are called skew. Two coplanar lines l and m are called parallel if either l = m or l ∩ m = ∅;otherwise,theyare called intersecting lines. Definition. Two planes α and β in W are called parallel if either α = β or α and β have no point in common. If the planes α and β are not parallel, then we say that they intersect each other. Basic Assumption 8 Two intersecting planes α and β meet in a straight line. Definition. Let l and m be two lines, and let C be an arbitrary point in 3-space. Draw the lines l and m passing through C and parallel to l and m, respectively. We say that l and m are perpendicular to each other if l and m are perpendicular to each other. Definition. Given a line l and a plane α in W ,wesaythatl is perpendicular to α if it is perpendicular to every line in α. Basic Assumption 9 For every point P and every plane α in W ,thereisa line l through P that is perpendicular to α. References 1. Bachmann, F., Aufbau der Geometrie aus dem Spiegelungsbegriff, Springer, Berlin, 1958; revised 2nd ed., 1973 2. Barnsley, M.F., Fractals Everywhere, Academic Press, Boston, 1988; revised 2nd ed., 1993 3. Berger, M., G´eom´etrie, vols. 1–5, CEDIC et Fernand Nathan, Paris, 1977; English translation Geometry I & II, Springer, Berlin, 1987 4. Bottema, O., Hoofdstukken uit de Elementaire Meetkunde, Servire, Den Haag 1944; revised 2nd ed., Epsilon Uitgaven, Utrecht, 1997; to be published in English by Springer, New York 5. Brauner, H., Lehrbuch der konstruktiven Geometrie, Springer, Wien, 1986 6. Brianchon, C.J. and J.V. Poncelet, G´eom´etrie des courves. Recherches sur la d´etermination d’une hyperbole ´equilat`ere, au moyen de quatres conditions donn´ees, Annales de Gergonne, 11, 1820–1821, 205–220 7. Brieskorn, E. and H. Knorrer¨ , Plane Algebraic Curves,Birkh¨auser, Basel, 1986 8. Cassels, J.W.S., An Introduction to the Geometry of Numbers, Springer, Berlin, 1997 (first published 1971) 9. Coxeter, H.S.M., Introduction to Geometry,2nd ed., Wiley, New York, 1969 10. Coxeter, H.S.M., Regular Polytopes,2nd edition, Macmillan, New York, 1963; revised edition, Dover, New York, 1973 11. Coxeter, H.S.M. and S.L. Greitzer, Geometry Revisited, New Mathematical Library 19, Mathematical Association of America, Washington, D.C., 1967 12. Coxeter, H.S.M., M. Emmer, R. Penrose, and M.L. Teuber,ed.,M.C. Es- cher, Arts and Science, Elsevier, North-Holland, Amsterdam, 1986 13. Cromwell, P.R, Polyhedra, Cambridge University Press, Cambridge, 1997 14. Descartes, The Geometry of Ren´e Descartes, with a Facsimile of the First Edition, translated by D. E. Smith and M. L. Latham, Open Court Publishing, 1925; reprinted Dover, New York, 1954 15. Descharnes, R. and G. Neret´ , Salvador Dal´ı, Edita, Lausanne, 1993 16. Devlin, K., Mathematics: The Science of Patterns, Scientific Americain Li- brary, HPHLP, New York, 1994 17. Ernst, B., Avonturen met onmogelijke figuren, Aramith, Amsterdam, 1985 18. Euclid, The Thirteen Books of the Elements, Vols. I, II, III, translated with introduction and commentary by T. L. Heath, Cambridge University Press, Cambridge, 1908; revised 2nd ed., Dover, New York, 1956 338 References 19. Feuerbach, K.W. Eigenschaften einiger merkw¨urdigen Punkte des geradlini- gen Dreiecks, Haarlem; 2nd ed., P. Visser Azn., 1908 20. Field, J.V., The Invention of Infinity, Mathematics and the Art in the Renais- sance, Oxford University Press, Oxford (USA), 1997 21. Goodstein, D.L. and J.R. Goodstein, Feynman’s Lost Lecture: The Motion of the Planets Around the Sun, Vintage, London, 1996 22. Greenberg, M.J., Euclidean and Non-Euclidean Geometries,3rd ed., Freeman, New York, 1993 23. Grunbaum,¨ B. and G.C. Shephard, Tilings and Patterns, Freeman, New York, 1987 24. Hahn, Liang-Shin, Complex Numbers and Geometry, Mathematical Associa- tion of America, Washington, D.C., 1994 25. Hardy, G.H. and E.M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, Oxford; 5th ed., 1980 26. Hartshorne, Robin, Geometry: Euclid and Beyond, Springer, New York, 2000 27. Heilbron, J.L., Geometry Civilized; History, Culture, and Technique,Claren- don Press, Oxford, 1998 28. Hofmann, K.H. and R. Wille,ed.,Symmetry of Discrete Mathematical Struc- tures and Their Symmetry Groups, A Collection of Essays, Research and Ex- position in Mathematics 15, Heldermann, Berlin, 1991 29. Hofstede, G., H.J. Meewis, Jac. Stolk, and C. Kros, Machineonderdelen, 18th edition, Morks, Dordrecht, 1970 30. Holden, A, Shapes, Space and Symmetry, Columbia University Press, New York, 1971; reprinted Dover, New York, 1991 31. Honsberger, R., Episodes in Nineteenth and Twentieth Century Euclidean Geometry, New Mathematical Library 37, Mathematical Association of Amer- ica, Washington, D.C., 1995 32. Hughes, B.B. Fibonacci’s De Practica Geometrie, Springer Verlag, New York, 2007 33. Huntley, H.E., The Divine Proportion, A Study in Mathematical Beauty, Dover, New York, 1970 34. Hutchinson, J.E., Fractals and Self Similarity, Indiana University Mathemat- ics Journal 30, 1981, 713–747 35. Jacobs, H.R., Geometry, Freeman, San Francisco, 1974 36. Kindt, M., Lessen in projectieve meetkunde,2nd ed., Epsilon, Utrecht, 1996 37. Kittel, C., Introduction to Solid State Physics,5th ed., Wiley, New York, 1976 38.
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  • The Secrets of Triangles: a Mathematical Journey

    The Secrets of Triangles: a Mathematical Journey

    2 3 Published 2012 by Prometheus Books The Secrets of Triangles: A Mathematical Journey. Copyright © 2012 by Alfred S. Posamentier and Ingmar Lehmann. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, digital, electronic, mechanical, photocopying, recording, or otherwise, or conveyed via the Internet or a website without prior written permission of the publisher, except in the case of brief quotations embodied in critical articles and reviews. Cover image © Media Bakery/Glenn Mitsui Jacket design by Jacqueline Nasso Cooke Inquiries should be addressed to Prometheus Books 59 John Glenn Drive Amherst, New York 14228–2119 OICE: 716–691–0133 FAX: 716–691–0137 WWW.PROMETHE SBOOKS.COM 16 15 14 13 12 5 4 3 2 1 Library of Congress Cataloging-in-Publication Data Posamentier, Alfred S. The secrets of triangles : a mathematical journey / by Alfred S. Posamentier and Ingmar Lehmann. p. cm. Includes bibliographical references and index. ISBN 978–1–61614–587–3 (cloth : alk. paper) ISBN 978–1–61614–588–0 (ebook) 1. Trigonometry. 2. Triangle. I. Lehmann, Ingmar. II. Title. QA531.P67 2012 516'.154—dc23 2012013635 4 Printed in the nited States of America on acid-free paper 5 6 Acknowledgments Preface 1. Introduction to the Triangle 2. Concurrencies of a Triangle 3. Noteworthy Points in a Triangle 4. Concurrent Circles of a Triangle 5. Special Lines of a Triangle 6. seful Triangle Theorems 7. Areas of and within Triangles 8. Triangle Constructions 9. Inequalities in a Triangle 10. Triangles and Fractals Appendix Notes References Index 7 The authors wish to extend sincere thanks for proofreading and useful suggestions to Dr.