Table Of ContentSotirios E. Louridas · Michael Th. Rassias
Problem-Solving
and Selected
Topics in Euclidean
Geometry
In the Spirit of the Mathematical
Olympiads
Foreword by Michael H. Freedman
Problem-Solving and Selected Topics
in Euclidean Geometry
Sotirios E. Louridas (cid:2) Michael Th. Rassias
Problem-Solving
and Selected
Topics in Euclidean
Geometry
In the Spirit of the Mathematical
Olympiads
Foreword by Michael H. Freedman
SotiriosE.Louridas MichaelTh.Rassias
Athens,Greece DepartmentofMathematics
ETHZurich
Zurich,Switzerland
ISBN978-1-4614-7272-8 ISBN978-1-4614-7273-5(eBook)
DOI10.1007/978-1-4614-7273-5
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013938266
MathematicsSubjectClassification: 51-XX,51-01
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Foreword
Geometry has apparently fallen on hard times. I learned from this excellent trea-
tise on plane geometry that U.S. President James A. Garfield constructed his own
proof of the Pythagorean Theorem in 1876, four years before being elected to an
unfortunatelybriefpresidency.
Inarecentlecture,ScottAaronson(MIT)offeredatongue-in-cheekanswertothe
question:“SupposethereisashortproofthatP (cid:2)=NP?”with,“Supposespacealiens
assassinated President Kennedy to prevent him from discovering such a proof?”
Ifounditpleasanttowonderwhichhalfofthetwoclauseswaslessprobable.Sadly
he concluded that it was more likely that space aliens were behind Kennedy’s as-
sassinationthanthatamodernpresidentwouldbedoingmathematics.Perhapsthis
bookoffershopethatwhatwaspossibleoncewillbepossibleagain.
Young people need such texts, grounded in our shared intellectual history and
challenging them to excel and create a continuity with the past. Geometry has
seemeddestinedtogivewayinourmoderncomputerizedworldtoalgebra.Aswith
MichaelTh.Rassias’previoushomonymousbookonnumbertheory,itisapleasure
toseethementaldisciplineoftheancientGreekssowellrepresentedtoayouthful
audience.
MicrosoftStationQ MichaelH.Freedman
CNSIBldg.,Office2245
UniversityofCalifornia
SantaBarbara,CA93106-6105
USA
v
Acknowledgements
We feel deeply honored and grateful to Professor Michael H. Freedman, who has
writtentheForewordofthebook.
We would like to express our thanks to Professors D. Andrica, M. Bencze,
S. Markatis, and N. Minculete for reading the manuscript and providing valuable
suggestions and comments which have helped to improve the presentation of the
book.WewouldalsoliketothankDr.A.Magkosforhisusefulremarks.
WewouldliketoexpressourappreciationtotheHellenicMathematicalSociety
for the excellentprogram of preparation of the young contestantsfor their partici-
pationinMathematicalOlympiads.
LastbutnotleastwewouldliketoofferourthankstoDr.F.I.Travlopanosforhis
invaluablehelpandforhisusefulsuggestions.
Finally,itisourpleasuretoacknowledgethesuperbassistanceprovidedbythe
staffofSpringerforthepublicationofthebook.
SotiriosE.Louridas
MichaelTh.Rassias
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 TheOriginofGeometry . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 AFewWordsAboutEuclid’sElements . . . . . . . . . . . . . . . 3
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 BasicConceptsofLogic . . . . . . . . . . . . . . . . . . . 5
2.1.2 OnRelatedPropositions . . . . . . . . . . . . . . . . . . . 7
2.1.3 OnNecessaryandSufficientConditions . . . . . . . . . . 8
2.2 MethodsofProof . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 ProofbyAnalysis . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 ProofbySynthesis . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 ProofbyContradiction . . . . . . . . . . . . . . . . . . . 13
2.2.4 MathematicalInduction . . . . . . . . . . . . . . . . . . . 15
3 FundamentalsonGeometricTransformations . . . . . . . . . . . . . 19
3.1 AFewFacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 ExamplesonTranslation . . . . . . . . . . . . . . . . . . 23
3.3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 SymmetrywithRespecttoaCenter . . . . . . . . . . . . . 25
3.3.2 SymmetrywithRespecttoanAxis . . . . . . . . . . . . . 25
3.4 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4.1 ExamplesofRotation . . . . . . . . . . . . . . . . . . . . 29
3.5 Homothety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5.1 ExamplesofHomothety . . . . . . . . . . . . . . . . . . . 36
3.6 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.6.1 InverseofaPoint . . . . . . . . . . . . . . . . . . . . . . 39
3.6.2 InverseofaFigure . . . . . . . . . . . . . . . . . . . . . . 39
3.6.3 AnInvarianceProperty . . . . . . . . . . . . . . . . . . . 40
3.6.4 BasicCriterion . . . . . . . . . . . . . . . . . . . . . . . . 40
ix
x Contents
3.6.5 AnotherInvarianceProperty . . . . . . . . . . . . . . . . . 41
3.6.6 InvertibilityandHomothety . . . . . . . . . . . . . . . . . 41
3.6.7 TangenttoaCurveandInversion . . . . . . . . . . . . . . 41
3.6.8 InversionandAngleofTwoCurves . . . . . . . . . . . . . 43
3.6.9 ComputingDistanceofPointsInversetoaThirdOne . . . 43
3.6.10 InverseofaLineNotPassingThroughaPole . . . . . . . 44
3.6.11 InverseofaCirclewithRespecttoaPoleNotBelonging
totheCircle . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6.12 InverseofaFigurePassingThroughthePoleofInversion . 46
3.6.13 OrthogonalCirclesandInversion . . . . . . . . . . . . . . 47
3.6.14 ApplicationsoftheInversionOperation . . . . . . . . . . . 48
3.7 TheIdeaBehindtheConstructionofaGeometricProblem . . . . . 51
4 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1 GeometricProblemswithBasicTheory . . . . . . . . . . . . . . . 79
5.2 GeometricProblemswithMoreAdvancedTheory . . . . . . . . . 82
5.3 GeometricInequalities . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1 GeometricProblemswithBasicTheory . . . . . . . . . . . . . . . 93
6.2 GeometricProblemswithMoreAdvancedTheory . . . . . . . . . 124
6.3 GeometricInequalities . . . . . . . . . . . . . . . . . . . . . . . . 187
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
A.1 TheGoldenSection . . . . . . . . . . . . . . . . . . . . . . . . . 215
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
IndexofSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
SubjectIndex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Chapter 1
Introduction
μημoυτoυςκυ´κλoυςτα´ραττ(cid:10).
(Donotdisturbmycircles.)
Archimedes(287BC–212BC)
In this chapter, we shall present an overview of Euclidean Geometry in a general,
non-technicalcontext.
1.1 TheOriginofGeometry
Generally, we could describe geometry as the mathematical study of the physical
worldthatsurroundsus,ifweconsiderittoextendindefinitely.Morespecifically,
we could define geometry as the mathematical investigation of the measure, the
propertiesandrelationshipsofpoints,lines,angles,shapes,surfaces,andsolids.
It is commonly accepted that basic methods of geometry were first discovered
andusedineverydaylifebytheEgyptiansandtheBabylonians.Itisremarkablethat
they could calculate simple areas and volumes and they had closely approximated
thevalueofπ (theratioofthecircumferencetothediameterofacircle).
However,eventhoughtheEgyptiansandtheBabylonianshadundoubtedlymas-
teredsomegeometricaltechniques,theyhadnotformedamathematicalsystemof
geometryasatheoreticalsciencecomprisingdefinitions,theorems,andproofs.This
wasinitiatedbytheGreeks,approximatelyduringtheseventhcenturyBC.
It is easy to intuitively understand the origin of the term geometry, if we ety-
mologicallystudythemeaningoftheterm.Thewordgeometryoriginatesfromthe
Greekwordγ(cid:10)ωμ(cid:10)τρι´α,whichisformedbytwootherGreekwords:Thewordγη,
whichmeansearthandthewordμ(cid:10)´τρoν,whichmeansmeasure.Hence,geometry
actually means the measurement of the earth, and originally, that is exactly what
itwasbeforetheGreeks.Forexample,inapproximately240BC,theGreekmath-
ematician Eratosthenes used basic but ingenious methods of geometry that were
developedtheoreticallybyseveralGreekmathematiciansbeforehistimeinorderto
measuretheEarth’scircumference.Itisworthmentioningthathesucceededtodo
so,withanerroroflessthan2%incomparisontotheexactlengthofthecircumfer-
enceasweknowittoday.Therefore,itisevidentthatgeometryarosefrompractical
activity.
S.E.Louridas,M.Th.Rassias,Problem-SolvingandSelectedTopicsin 1
EuclideanGeometry,DOI10.1007/978-1-4614-7273-5_1,
©SpringerScience+BusinessMediaNewYork2013
2 1 Introduction
Geometrywasdevelopedgraduallyasanabstracttheoreticalsciencebymathe-
maticians/philosophers,suchasThales,Pythagoras,Plato,Apollonius,Euclid,and
others.Morespecifically,Thales,apartfromhisintercepttheorem,isalsothefirst
mathematiciantowhomtheconceptofproofbyinductionisattributed.Moreover,
PythagorascreatedaschoolknownasthePythagoreans,whodiscoverednumerous
theoremsingeometry.Pythagorasissaidtobethefirsttohaveprovidedadeductive
proofofwhatisknownasthePythagoreanTheorem.
Theorem1.1(PythagoreanTheorem) Inanyrighttrianglewithsidesoflengthsa,
b,c,wherecisthelengthofthehypotenuse,itholds
a2+b2=c2. (1.1)
Theabovetheoremhascapturedtheinterestofbothgeometersandnumbertheo-
ristsforthousandsofyears.Hundredsofproofshavebeenpresentedsincethetime
of Pythagoras.It is amusingto mentionthat eventhe 20thpresident of theUnited
States, J.A. Garfield, was so much interested in this theorem that he managed to
discoveraproofofhisown,in1876.
The number theoretic aspect of the Pythagorean Theorem is the study of the
integervalues a, b, c,whichsatisfyEq.(1.1).Suchtriplesofintegers (a,b,c) are
called Pythagorean triples [86]. Mathematicians showed a great interest in such
propertiesofintegersandwereeventuallyleadtotheinvestigationofthesolvability
ofequationsoftheform
an+bn=cn,
wherea,b,c∈Z+ andn∈N,n>2.
ThesestudiesleadafterhundredsofyearstoWiles’celebratedproofofFermat’s
LastTheorem[99],in1995.
Theorem1.2(Fermat’sLastTheorem) Itholds
an+bn(cid:2)=cn,
foreverya,b,c∈Z+ andn∈N,n>2.
Let us now go back to the origins of geometry. The first rigorous foundation
which made this discipline a well-formed mathematical system was provided in
Euclid’sElementsinapproximately300BC.
The Elements are such a unique mathematical treatise that there was no need
foranykindofadditionsormodificationsformorethan2000years,untilthetime
of the great Russian mathematician N.I. Lobac˘evski˘i (1792–1856) who developed
anewtypeofgeometry,knownashyperbolicgeometry,inwhichEuclid’sparallel
postulatewasnotconsidered.
