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1、AnEquilateralDilemmaWeclosedthestatmentofourAugustproblemwiththefollowing:Comment:Wedidnotseeaneasysolutiontothisproblem.Asyoureadoursolution,you'llseewhatwemeant.Mostofthesubmitedsolutionsdemonstratedthefundamentalchallengeofthisproblem:Howdoyouprovesomethingthatseemsobviouslytrue?Thep
2、roblem'ssimplicityandtheintuitiveobviousness"oftheconclusionleadmanyrespondersdowndeadendpaths.SomesuggestedusingtrigonometricfunctionstoprovetriangleABCequilateral,butnoneoftheothertrianglespresentareimplicitlyright"triangles.OthersbeganwiththeassumptionthattrianglesADF,BED,andCFEwer
3、econgruent,whichcannotbeas-sumed,butratherneedstobeproved.Hereisoursolution.Wearestillinterestedinamoreelegantsolutionifanyonewishestosubmitone!Weproceedbybreakingintocases.LetthesidesofDEFhavelengthx,andthelinesegmentsAD,BE,andCFhavelengthy.Theformofourysolutionwilldependonthevalueof.W
4、ebeginwithsomespecialcases.xy2Case1,when=px3InthiscasethelargestvaluepossibleforangleAwouldbe60{whenangleADF=90(seediagram).SinceAD=BE=CF,thisholdstruefortrianglesBEDandCFEaswell.Inthiscasethen,sinceanglesA,B,andCmustsumto180,theymustallbe60,andABCequilateral.y2Case2,when>px3Inthisc
5、ase,asxgrowssmaller(orylarger,orboth).AngleAonlygrowssmaller(seeabovediagram).Again,sincethisisoccuringequallythroughouttriangleABC(AD=BE=CF,andtriangleDEFisequilateral),anglesA,B,andCmustbeunder60,whichmakesitverydicultforthemtosumto180.Socase2isnotpossible.yCase3,when=1,orwhentrian
6、gleADFisisosceles.x1Forconvenience,labletheanglesasfollows:ADF=,FAD=A,DFA=,CFE=,andBED=ϵ.Weknowthat=A=90 :Thisimplies2that=180 (+60).Substituting(90 ())for,weget=()+30.22Similarly,ϵ=()+30,and=(ϵ)+30.Thesystemoflinearequations22hastheuniquesolution=A===ϵ=60.InparticularA=60
7、,andthesameargumentimpliesthatallthreeanglesofthelargetriangleare60anditisthereforeequilateral.yCase4,when0<<1xConsider xedvaluesofx;ywith0
