Geometric problems on maxima and minima

272 Pages·2006·2.967 MB·English

by  Titu Andreescu, Oleg Mushkarov, Luchezar Stoyanov

Titu Andreescu
Oleg Mushkarov
Luchezar Stoyanov
Geometric Problems
on Maxima and Minima
Birkha¨user
Boston • Basel • Berlin
TituAndreescu OlegMushkarov
TheUniversityofTexasatDallas BulgarianAcademyofSciences
DepartmentofScience/ InstituteofMathematicsandInformatics
MathematicsEducation 1113Sofia
Richardson,TX75083 Bulgaria
USA
LuchezarStoyanov
TheUniversityofWesternAustralia
SchoolofMathematicsandStatistics
Crawley,PerthWA6009
Australia
CoverdesignbyMaryBurgess.
MathematicsSubjectClassification(2000):00A07,00A05,00A06
LibraryofCongressControlNumber:2005935987
ISBN-100-8176-3517-3 eISBN0-8176-4473-3
ISBN-13978-0-8176-3517-6
Printedonacid-freepaper.
(cid:2)c2006Birkha¨userBoston
BasedontheoriginalBulgarianedition,Ekstremalnizadachivgeometriata,
NarodnaProsveta,Sofia,1989
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Contents
Preface vii
1 MethodsforFindingGeometricExtrema 1
1.1 EmployingGeometricTransformations . . . . . . . . . . . . . . 1
1.2 EmployingAlgebraicInequalities . . . . . . . . . . . . . . . . . 19
1.3 EmployingCalculus . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4 TheMethodofPartialVariation . . . . . . . . . . . . . . . . . . 38
1.5 TheTangencyPrinciple . . . . . . . . . . . . . . . . . . . . . . . 48
2 SelectedTypesofGeometricExtremumProblems 63
2.1 Isoperimetric Problems . . . . . . . . . . . . . . . . . . . . . . . 63
2.2 ExtremalPointsinTriangleandTetrahedron . . . . . . . . . . . . 72
2.3 Malfatti’s Problems . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.4 ExtremalCombinatorial GeometryProblems . . . . . . . . . . . 88
3 Miscellaneous 95
3.1 TriangleInequality . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2 SelectedGeometricInequalities . . . . . . . . . . . . . . . . . . 96
3.3 MaxMinandMinMax. . . . . . . . . . . . . . . . . . . . . . . . 98
3.4 AreaandPerimeter . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.5 PolygonsinaSquare . . . . . . . . . . . . . . . . . . . . . . . . 101
3.6 BrokenLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.7 Distribution ofPoints . . . . . . . . . . . . . . . . . . . . . . . . 102
3.8 Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4 HintsandSolutionstotheExercises 105
4.1 EmployingGeometricTransformations . . . . . . . . . . . . . . 105
4.2 EmployingAlgebraicInequalities . . . . . . . . . . . . . . . . . 124
4.3 EmployingCalculus . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.4 TheMethodofPartialVariation . . . . . . . . . . . . . . . . . . 151
vi Contents
4.5 TheTangencyPrinciple . . . . . . . . . . . . . . . . . . . . . . . 161
4.6 Isoperimetric Problems . . . . . . . . . . . . . . . . . . . . . . . 169
4.7 ExtremalPointsinTriangleandTetrahedron . . . . . . . . . . . . 176
4.8 Malfatti’s Problems . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.9 ExtremalCombinatorial GeometryProblems . . . . . . . . . . . 188
4.10 TriangleInequality . . . . . . . . . . . . . . . . . . . . . . . . . 197
4.11 SelectedGeometricInequalities . . . . . . . . . . . . . . . . . . 200
4.12 MaxMinandMinMax. . . . . . . . . . . . . . . . . . . . . . . . 212
4.13 AreaandPerimeter . . . . . . . . . . . . . . . . . . . . . . . . . 215
4.14 PolygonsinaSquare . . . . . . . . . . . . . . . . . . . . . . . . 233
4.15 BrokenLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
4.16 Distribution ofPoints . . . . . . . . . . . . . . . . . . . . . . . . 240
4.17 Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Notation 255
GlossaryofTerms 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
frequentlymet. Notsurprisingly, peoplehavebeendealingwithsuchproblemsfor
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
toproviderigorous proofs.
Forexample,onemightmention hereHeron’s(firstcentury CE)discovery that
the light ray in space incoming from a point A and outgoing through a point B
after reflection atamirror α travels the shortest possible path from A to B having
acommonpointwithα.
Another famous problem, the so-called isoperimetric problem, wasconsidered
for example by Descartes (1596–1650): Of all plane figures with a given perime-
ter, 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 rig-
orous 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 narrowstrips, she madealong string withwhichshe wassupposed to
surround as large as possible area on the seashore. How to do this in an optimal
wayisaproblemcloselyrelatedtothepreviousone,andinfactasolutioniseasily
foundonceoneknowsthemaximizingpropertyofthecircle.
Another problem that is both interesting and easy to state was posed in 1775
byI.F.Fagnano: Inscribe atriangle ofminimal perimeter in agiven acute-angled
triangle. Anelegantsolutiontothisrelativelysimple“networkproblem”wasgiven
byHermannSchwarz(1843–1921).
viii Preface
Most of these classical problems are discussed in Chapter 1, which presents
severaldifferentmethodsforsolvinggeometric problemsonmaximaandminima.
One of these concerns applications of geometric transformations, e.g., reflection
through aline or plane, rotation. Thesecond is about appropriate use of inequali-
ties. Another analytic method is the application of tools from the differential cal-
culus. ThelasttwomethodsconsideredinChapter1aremoregeometricinnature;
these are the method of partial variation and the tangency principle. Their names
speakforthemselves.
Chapter 2 is devoted to several types of geometric problems on maxima and
minimathat are frequently met. Herefor example wediscuss avariety of isoperi-
metric problems similar in nature to the ones mentioned above. Various distin-
guishedpointsinthetriangleandthetetrahedron canbedescribedasthesolutions
ofsomespecificproblems onmaximaorminima. Section 2.2considers examples
of this kind. An interesting type of problem, called Malfatti’s problems, are con-
tainedinSection2.3;theseconcernthepositioningofseveraldisksinagivenfigure
intheplanesothatthesumoftheareas ofthedisksismaximal. Section2.4deals
withsomeproblemsonmaximaandminimaarisingincombinatorial geometry.
Chapter3collectssomegeometricproblemsonmaximaandminimathatcould
not be put into any of the firsttwo chapters. Finally, Chapter 4provides solutions
andhintstoallproblemsconsidered inthefirstthreechapters.
Each section in the book is augmented by exercises and more solid problems
forindividual work. Tomakeiteasier tofollow the arguments inthe book alarge
numberoffiguresisprovided.
ThepresentbookispartlybasedonitsBulgarianversionExtremalProblemsin
Geometry, written by O. Mushkarov and L. Stoyanov and published in 1989 (see
[16]). Thisnewversionretainsabouthalfofthecontentsoftheoldone.
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
injournalarticlesorusedinvariousmathematicscompetitionsaroundtheworld—
thewholeexposition ofthebookiskeptatasufficientlyelementary levelsothatit
canbeunderstood byhigh-school students.
Apartfromtryingtobecomprehensive intermsoftypesofproblemsandtech-
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
achievedthistoareasonable 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
differentpartsoftheworldformorethan25years. Theauthorshopethatthebook
Preface ix
will appeal to a wide audience of high-school students and mathematics teachers,
graduate students, professional mathematicians, and puzzle enthusiasts. Thebook
willbeparticularlyusefultostudentsinvolvedinmathematicscompetitionsaround
theworld.
WearegratefultoSvetoslavSavchevandNevenaSabevaforhelpingusduring
thepreparationofthisbook,andtoDavidKramerforthecorrectionsandimprove-
mentshemadewhenediting thetextforpublication.
TituAndreescu
OlegMushkarov
LuchezarStoyanov
September,2005
Geometric Problems
on Maxima and Minima
Chapter 1
Methods for Finding Geometric
Extrema
1.1 Employing Geometric Transformations
Itisarathercommonfeatureinsolvinggeometricproblemsthattheobjectofstudy
undergoessomegeometrictransformationinorderforittobebroughttoasituation
thatiseasiertodealwith. Inthepresentsectionthismethodisusedtosolvecertain
geometricproblemsonmaximaandminima. Thetransformations involvedarethe
well-knownsymmetrywithrespecttoalineorapoint,rotation,anddilation. Apart
fromthis,insomespacegeometryproblemswearegoingtousesymmetrythrough
aplane,rotationaboutaline,andspacedilation. Wereferthereaderto[17]or[22]
forgeneral information aboutgeometric transformations.
WebeginwiththewellknownHeron’sproblem.
Problem1.1.1 Aline (cid:3)isgiven intheplane and twopoints Aand B lying onthe
same side of (cid:3). Find a point X on (cid:3) such that the broken line AXB has minimal
length.
Solution. Let B(cid:2)bethereflectionof Bin(cid:3)(Fig.1). Bythepropertiesofsymmetry,
wehave XB = XB(cid:2) foranypoint X on(cid:3),so
AX + XB = AX + XB(cid:2) ≥ AB(cid:2).
Theequality occursprecisely when X istheintersection point X of(cid:3)andtheline
0
segment AB(cid:2). Thus,foranypoint X on(cid:3)different from X ,
0
AX + XB ≥ AB(cid:2) = AX + X B,
0 0
whichshowsthat X istheuniquesolution oftheproblem. ♠
0