Geometric Problems on Maxima and Minima

Questions of maxima and minima have great practical significance, with applications to physics, engineering, and economics; they have also given rise to theoretical advances, notably in calculus and optimization. Indeed, while most texts view the study of

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on M A X I M A and M I N I M A

Ti t u A n d r e e s c u O l e g M u s h k a r ov L u c h e z a r Stoya n ov Birkhäuser

Titu Andreescu Oleg Mushkarov Luchezar Stoyanov

Geometric Problems on Maxima and Minima

Birkh¨auser Boston • Basel • Berlin

Titu Andreescu The University of Texas at Dallas Department of Science/ Mathematics Education Richardson, TX 75083 USA

Oleg Mushkarov Bulgarian Academy of Sciences Institute of Mathematics and Informatics 1113 Sofia Bulgaria

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 ISBN-13 978-0-8176-3517-6

eISBN 0-8176-4473-3

Printed on acid-free paper. c 2006 Birkh¨auser Boston 

Based on the original Bulgarian edition, Ekstremalni zadachi v geometriata, Narodna Prosveta, Sofia, 1989 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA) and the author, 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 in the United States of America. 987654321 www.birkhauser.com

(KeS/MP)

Contents Preface

vii

1 Methods for Finding Geometric Extrema 1.1 Employing Geometric Transformations 1.2 Employing Algebraic Inequalities . . . 1.3 Employing Calculus . . . . . . . . . . . 1.4 The Method of Partial Variation . . . . 1.5 The Tangency Principle . . . . . . . . .

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1 1 19 27 38 48

2 Selected Types of Geometric Extremum Problems 2.1 Isoperimetric Problems . . . . . . . . . . . . . 2.2 Extremal Points in Triangle and Tetrahedron . . 2.3 Malfatti’s Problems . . . . . . . . . . . . . . . 2.4 Extremal Combinatorial Geometry Problems .

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63 63 72 80 88

3 Miscellaneous 3.1 Triangle Inequality . . . . . . . 3.2 Selected Geometric Inequalities 3.3 MaxMin and MinMax . . . . . . 3.4 Area and Perimeter . . . . . . . 3.5 Polygons in a Square . . . . . . 3.6 Broken Lines . . . . . . . . . . 3.7 Distribution of Points . . . . . . 3.8 Coverings . . . . . . . . . . . .

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