On an area property of the sum cotA+cotB+cotC in a

OnanareapropertyofthesumcotA+cotB+cotΓinatriangle1.IntroductionRummagingthroughanobscuretrigonometrybookpublishedinAthens,Greece(andintheGreeklanguage),andlongoutofprint,Idiscoveredthefollowing,listedasanexercise(seereference[1]):Quote:“LetΓΒΑΔbeatriangle.IfonthesideABwedrawthelineperpendiculartoABatthepointB;onthesideBΓthelineperpendiculartoitatthepointΓ;andonΓAtheperpendiculartoitatA;anewtriangleA΄Β΄Γ΄isformed.ProvethatEE΄=(cotΑ+cotΒ+cotΓ)2,whereE΄andEstandfortheareasofthetrianglesΑ΄ΒΑΔ΄Γ΄andΓΒΑΔrespectively”.endofquote1ImportantNoteRegardingAngleNotationIntheinterestofclarityandavoidanceofconfusiononthepartofthereader,pleasenote:1)ByreferingtotheanglesA,B,Γ,wealwaysbereferingtotheinternaltrianglesA,B,Γ,inthecontextofaclearlydefinedtriangleΓΒΑΔ.Inthatcontext,angleA=BAΓ=ΓAB;orbyusingthealternativenotation^,thealternativenotation^,angleA=BAΓ=ΓAB.^^Similarly,angleB=ABΓ=ΓBAand^^angleΓ=AΓB=BΓA.Consequently,allsixanglesA,B,Γ,A’,B’,Γ’,referredtoinFigures1through4,aretheinternalanglesofthetrianglesABΓandA’B’Γ’.2)Inanyothercontext,wewoulduseanyofthetwostandardthree-letteranglenotations(mentionedabove),withthevertexlettersituatedinthemiddle.However,inthispaper,such“othercontext”,doesnottrulyarise.3)Inapaperlikethis,wheretheinternalanglesA,B,Γarerepeatedveryfrequentlythroughout,usingathree-letternotation(insteadofasingleletterone),wouldconceivablyresultinasignificantincreaseinthelengthofthepaper.2AfirstobservationrevealsthatthesumcotΑ+cotΒ+cotΓcanbearbitrarilylarge.Indeed,ifwerestrictourattentiononlytothosetrianglesinwhichnoneoftheanglesA,B,Γisobtuse;theneachofthetrigonometricnumberscotΑ,cotΒ,cotΓisboundedbelowbyzero,i.e.cotA,cotB,cotΓ≥0.Butsay,angleΓcanbecomearbitrarilysmall;inthelanguageofcalculus,Γ→0+;sothatcotΓ→+∞.AsecondobservationshowsthatthetrianglesΓΒΑΔandΓ΄΄Β΄ΑΔarealwayssimilar(seeFigures2,3,4).FurthermorenotethatifweweretoeffectasimilarconstructionoftriangleΓ΄΄Β΄ΑΔfromagiventriangleΓΒΑΔbutusinganangleϕ,0°ϕ≤90°(insteadofjustϕ=90°),thesamefactwouldemerge;thatis,thetwotrianglesaresimilar(seeFigure1).Thispaperhasatwo-foldaim:firstestablishthatEE΄=(cotΑ+cotΒ+cotΓ)2;andsecondlythattheminimumvalueofthisarearatiois3,attainedpreciselywhenthetriangleΓΒΑΔisequilateral(andthus,bysimilarity,triangleΓ΄΄Β΄ΑΔaswell).Eventhough,asweshallsee,amongalltrianglepairs(ΓΒΑΔ,Γ΄΄Β΄ΑΔ),theoneswiththesmallestratio(=3)arethosepairsinwhichbothtrianglesareequilateral;ifwerestrictoursearchonlytothosepairsinwhichbothΓΒΑΔandΓ΄΄Β΄ΑΔarerighttriangles,then,aswewillshow,theminimumvalueoftheabovearearatioisequalto4;obtainedpreciselywhenbothrighttrianglesareisosceles.2.Illustrations343.AfewpreliminariesThenineformulaslistedbelowpertaintoatrianglewithsidelengthsα=ΒΓ,β=ΓΑ,γ=ΑΒ,anglesΑ,Β,Γ,semi-perimeterr(i.e.2r=α+β+γ)andareaE.Ofthesenineformulas,thefirstfivearewellknowntoawidemathematicalaudience,whilethelastfour(Formulas6,7,8,9)arelessso;foreachofthoseformulas,Formulas6,7,8,and9weofferashortproofinthenextsection.Formula1(Heron’sFormula):E=γ)-(rβ)-(rα)-(rrFormula2:cos2θ=2cos2θ-1=1-2sin2θforanyangleθ,typicallymeasuredindegreesorradiansFormula3(LawofCosines):α2=β2+γ2-2βγcosA,β2=α2+γ2-2αγcosB,γ2=α2+β2-2αβcosΓFormula4(summationformulas):Foranyanglesθandω,cos(θ+ω)=cosθcosω-sinθsinω,sin(θ+ω)=sinθcosω+cosθsinωFormula5:E=21βγsinA=21αβsinΓ=21γαsinΒFormula6:16E2=2(α2β2+β2γ2+γ2α2)-(α4+β4+γ4)Formula7:Foranyangleθwhichisnottheformκπorκπ+2π,whereκisaninteger,cot2θ=2cotθ1-θcot25Formula8:cot2A=⎟⎠⎞⎜⎝⎛γ)-(rβ)-(rα)-(rr,cot2B=⎟⎠⎞⎜⎝⎛γ)-(rα)-(rβ)-(rr,cot2Γ=⎟⎠⎞⎜⎝⎛α)-(rβ)-(rγ)-(rrFormula9:E=222αβγ4(cotΑcotBcotΓ)++++4.ProofsofFormulas6,7,8,and9a)ProofofFormula6:FromFormula1wehave,E=γ)-(rβ)-(rα)-(rr⇔16E2=16r;γ)-(rβ)-(rα)-(r16E2=⎟⎠⎞⎜⎝⎛++2γβα⎟⎠⎞⎜⎝⎛+2α-γβ⎟⎠⎞⎜⎝⎛+2β-γα⎟⎠⎞⎜⎝⎛+2γ-βα⋅16=(α+β+γ)(β+γ-α)[α-(β-γ)](α+β-γ)=[(β+γ)2-α2)][α2-(β-γ)2]=α2(β+γ)2-α4-[(β+γ)(β-γ)]2+α2(β-γ)2=α2[(β+γ)2+(β-γ)2]-α4-[(β2-γ2)]2=α2[2(β2+γ2)]-α4-β4-γ4+2β2γ2=2(α2β2+β2γ2+γ2α2)-(α4+β4+γ4)‰b)ProofofFormula7:Foranytwoanglesθandωsuchthatθ+ω≠κπ,κanyinteger(sothatsin(ω+θ)≠0)wehave,inaccordancewithFormula4,cot(θ+ω)=ω)(θsinω)(θcos++=cosθcosω-sinθsinωsinθcosω+sinωcosθ6If,inadditiontotheabovecondition,wealsohaveθ≠mπandω≠nπ(wherem,nareanyintegers),thensinθsinω≠0;andso,cos(θ+ω)=cosθcosωsinθsinω-sinθsinωsinθsinωsinθcosωsinωcosθ-sinθsinωsinωsinθ=cotθcotω-1cotω+cotθThus,bysettingθ=ωweobtaincot2θ=2cotθ1-θcot2,undertheconditionsθ≠πand2θ≠π;anyinteger.Consideringthecasesevenandodd,theconditioncanbesimplifiedintoθ≠κπandθ≠κπ+lllll2π,κanyinteger.‰c)ProofofFormula8:Weneedonlyestablishthefirstofthethree(sub)formulas,theothertwoareestablishedsimilarly.Indeed,ifweapplythefirstofthetwo(sub)formulasinFormula2withθ=2A;incombinationwiththefirst(sub)formulainFormula3weobtain,cos2⎟⎠⎞⎜⎝⎛2A=2βγ2α-γβ1222++⇔cos2⎟⎠⎞⎜⎝⎛2A=βγ4α-γ)(β22+⇔cos2⎟⎠⎞⎜⎝⎛2A=βγ4α)γ(βα)-γ(β+++=βγ4(2r)α)-(r2=βγrα)-(r(1)Likewise,weapplycos2θ=1-2sin2θwithθ=2Ainconjunctionwiththefirst(sub)formulainFormula3toobtain,sin2⎟⎠⎞⎜⎝⎛2A=2βγ2α-γβ-1222⎟⎟⎠⎞⎜⎜⎝⎛+=βγ4γ)-(β-α227⇔sin2⎟⎠⎞⎜⎝⎛2A=βγ4γ)-β(αβ)-γ(α++=βγ4γ)-(r2β)-(r2=βγγ)-(rβ)-(r(2)Equations(1)and(2)implycot2⎟⎠⎞⎜⎝⎛2A=()()2Asin2A