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1、ForumGeometricorumbVolume7(2007)99102.bbbFORUMGEOMISSN1534-1178AVisualProofoftheErdos-MordellInequality˝ClaudiAlsinaandRogerB.NelsenAbstract.WepresentavisualproofofalemmathatreducestheproofoftheErd˝os-Mordellinequalitytoelementaryalgebra.In1935,thefollowingpro
2、blemproposalappearedintheAdvancedProblemssectionoftheAmericanMathematicalMonthly[5]:3740.ProposedbyPaulErdos,TheUniversity,Manchester,England˝.FromapointOinsideagiventriangleABCtheperpendicularsOP,OQ,ORaredrawntoitssides.ProvethatOA+OB+OC≥2(OP+OQ+OR).Trigonome
3、tricsolutionsbyMordellandBarrowappearedin[11].Theproofs,however,werenotelementary.Infact,nosimpleandelementaryproofofwhathadbecomeknownastheErd˝os-Mordelltheoremwasknownaslateas1956[13].Sincethenavarietyofproofshaveappeared,eachoneinsomesensesimplerormoreeleme
4、ntarythantheprecedingones.In1957Kazarinoffpublishedaproof[7]baseduponatheoreminPappusofAlexandriasMathematicalCollection;andayearlaterBankoffpublishedaproof[2]usingorthogonalprojectionsandsimilartriangles.Proofsusingareainequalitiesappearedin1997and2004[4,9].P
5、roofsemployingPtolemystheoremappearedin1993and2001[1,10].Atrigonometricproofofageneralizationoftheinequalityin2001[3],subsequentlygeneralizedin2004[6].Manyoftheseauthorsspeakglowinglyofthisresult,referringtoitasabeautifulinequality[9],aremarkableinequality[12]
6、,thefamousErd˝os-Mordellinequality[4,6,10],andthecelebratedErd˝os-Mordellinequality...abeautifulpieceofelementarymathematics[3].Inthisshortnotewecontinuetheprogressiontowardssimplerproofs.FirstwepresentavisualproofofalemmathatreducestheproofoftheErd˝os-Mordell
7、inequalitytoelementaryalgebra.ThelemmaprovidesthreeinequalitiesrelatingthelengthsofthesidesofABCandthedistancesfromOtotheverticesandtothesides.Whiletheinequalitiesinthelemmaarenotnew,webelieveourproofofthelemmais.Theproofusesnothingmoresophisticatedthanelement
8、arypropertiesoftriangles.InFigure1(a)weseethetriangleasdescribedbyErd˝os,andinFigurePublicationDate:April30,2007.CommunicatingEditor:PaulYiu.100C.AlsinaandR.B.Nelsen1(b)wedenotethe
