Geometric Problems on Maxima and Minima

Titu Andreescu Oleg Mushkarov Luchezar Stoyanov Geometric Problems on Maxima and Minima Birkhauser¨ Boston • Basel • Berlin Titu Andreescu Oleg Mushkarov The University of Texas at Dallas Bulgarian Academy of Sciences Department of Science/ Institute of Mathematics and Informatics Mathematics Education 1113 Sofia Richardson, TX 75083 Bulgaria USA Luchezar Stoyanov The University of Western Australia School of Mathematics and Statistics Crawley, Perth WA 6009 Australia Cover design by Mary Burgess. Mathematics Subject Classification (2000): 00A07, 00A05, 00A06 Library of Congress Control Number: 2005935987 ISBN-10 0-8176-3517-3 eISBN 0-8176-4473-3 ISBN-13 978-0-8176-3517-6 Printed on acid-free paper. c 2006 Birkhauser¨ Boston Based on the original Bulgarian edition, Ekstremalni zadachi v geometriata, Narodna Prosveta, Sofia, 1989 All rights reserved. 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(KeS/MP) 987654321 www.birkhauser.com Contents Preface vii 1 Methods for Finding Geometric Extrema 1 1.1 Employing Geometric Transformations . ............ 1 1.2 Employing Algebraic Inequalities ................. 19 1.3EmployingCalculus......................... 27 1.4 The Method of Partial Variation .................. 38 1.5TheTangencyPrinciple....................... 48 2 Selected Types of Geometric Extremum Problems 63 2.1 Isoperimetric Problems ....................... 63 2.2ExtremalPointsinTriangleandTetrahedron............ 72 2.3 Malfatti’s Problems . ....................... 80 2.4ExtremalCombinatorialGeometryProblems........... 88 3 Miscellaneous 95 3.1 Triangle Inequality . ....................... 95 3.2 Selected Geometric Inequalities .................. 96 3.3MaxMinandMinMax........................ 98 3.4AreaandPerimeter......................... 99 3.5 Polygons in a Square . ....................... 101 3.6BrokenLines............................ 101 3.7 Distribution of Points . ....................... 102 3.8Coverings.............................. 104 4 Hints and Solutions to the Exercises 105 4.1 Employing Geometric Transformations . ............ 105 4.2 Employing Algebraic Inequalities ................. 124 4.3EmployingCalculus......................... 136 4.4 The Method of Partial Variation .................. 151 vi Contents 4.5TheTangencyPrinciple....................... 161 4.6 Isoperimetric Problems ....................... 169 4.7ExtremalPointsinTriangleandTetrahedron............ 176 4.8 Malfatti’s Problems . ....................... 185 4.9ExtremalCombinatorialGeometryProblems........... 188 4.10 Triangle Inequality . ....................... 197 4.11 Selected Geometric Inequalities .................. 200 4.12MaxMinandMinMax........................ 212 4.13AreaandPerimeter......................... 215 4.14 Polygons in a Square . ....................... 233 4.15BrokenLines............................ 237 4.16 Distribution of Points . ....................... 240 4.17Coverings.............................. 250 Notation 255 Glossary of Terms 257 Bibliography 263 Preface Problems on maxima and minima arise naturally not only in science and engi- neering and their applications but also in daily life. A great variety of these have geometric nature: finding the shortest path between two objects satisfying certain conditions or a figure of minimal perimeter, area, or volume is a type of problem frequently met. Not surprisingly, people have been dealing with such problems for a very long time. Some of them, now regarded as famous, were dealt with by the ancient Greeks, whose intuition allowed them to discover the solutions of these problems even though for many of them they did not have the mathematical tools to provide rigorous proofs. For example, one might mention here Heron’s (first century CE) discovery that the light ray in space incoming from a point A and outgoing through a point B after reflection at a mirror α travels the shortest possible path from A to B having a common point with α. Another famous problem, the so-called isoperimetric problem, was considered for example by Descartes (1596–1650): Of all plane figures with a given perimeter, find the one with greatest area. That the “perfect figure” solving the problem is the circle was known to Descartes (and possibly much earlier); however, a rigorous proof that this is indeed the solution was first given by Jacob Steiner in the nineteenth century. A slightly different isoperimetric problem is attributed to Dido, the legendary queen of Carthage. She was allowed by the natives to purchase a piece of land on the coast of Africa “not larger than what an oxhide can surround.” Cutting the oxhide into narrow strips, she made a long string with which she was supposed to surround as large as possible area on the seashore. How to do this in an optimal way is a problem closely related to the previous one, and in fact a solution is easily found once one knows the maximizing property of the circle. Another problem that is both interesting and easy to state was posed in 1775 by I. F. Fagnano: Inscribe a triangle of minimal perimeter in a given acute-angled triangle. An elegant solution to this relatively simple “network problem” was given by Hermann Schwarz (1843–1921). viii Preface Most of these classical problems are discussed in Chapter 1, which presents several different methods for solving geometric problems on maxima and minima. One of these concerns applications of geometric transformations, e.g., reflection through a line or plane, rotation. The second is about appropriate use of inequalities. Another analytic method is the application of tools from the differential cal- culus. The last two methods considered in Chapter 1 are more geometric in nature; these are the method of partial variation and the tangency principle. Their names speak for themselves. Chapter 2 is devoted to several types of geometric problems on maxima and minima that are frequently met. Here for example we discuss a variety of isoperimetric problems similar in nature to the ones mentioned above. Various distin- guished points in the triangle and the tetrahedron can be described as the solutions of some specific problems on maxima or minima. Section 2.2 considers examples of this kind. An interesting type of problem, called Malfatti’s problems, are con- tained in Section 2.3; these concern the positioning of several disks in a given figure in the plane so that the sum of the areas of the disks is maximal. Section 2.4 deals with some problems on maxima and minima arising in combinatorial geometry. Chapter 3 collects some geometric problems on maxima and minima that could not be put into any of the first two chapters. Finally, Chapter 4 provides solutions and hints to all problems considered in the first three chapters. Each section in the book is augmented by exercises and more solid problems for individual work. To make it easier to follow the arguments in the book a large number of figures is provided. The present book is partly based on its Bulgarian version Extremal Problems in Geometry, written by O. Mushkarov and L. Stoyanov and published in 1989 (see [16]). This new version retains about half of the contents of the old one. Altogether the book contains hundreds of geometric problems on maxima or minima. Despite the great variety of problems considered—from very old and classical ones like the ones mentioned above to problems discussed very recently in journal articles or used in various mathematics competitions around the world— the whole exposition of the book is kept at a sufficiently elementary level so that it can be understood by high-school students. Apart from trying to be comprehensive in terms of types of problems and tech- niques for their solutions, we have also tried to offer various different levels of difficulty, thus making the book possible to use by people with different interests in mathematics, different abilities, and of different age groups. We hope we have achieved this to a reasonable extent. The book reflects the experience of the authors as university teachers and as people who have been deeply involved in various mathematics competitions in different parts of the world for more than 25 years. The authors hope that the book Preface ix will appeal to a wide audience of high-school students and mathematics teachers, graduate students, professional mathematicians, and puzzle enthusiasts. The book will be particularly useful to students involved in mathematics competitions around the world. We are grateful to Svetoslav Savchev and Nevena Sabeva for helping us during the preparation of this book, and to David Kramer for the corrections and improve- ments he made when editing the text for publication. Titu Andreescu Oleg Mushkarov Luchezar Stoyanov September, 2005 Geometric Problems on Maxima and Minima Chapter 1 Methods for Finding Geometric Extrema 1.1 Employing

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