Prove-that-if-the-lengths-of-a-triangle-form-an-arithmetic-progression-then-the-centre-of-incircle-and-the-centroid-of-triangle-lie-on-a-line-parallel-to-the-side-of-middle-length-of-the-triangle-

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Question Number 62380 by ajfour last updated on 20/Jun/19

$$Provethatifthelengthsofa$$$$triangleformanarithmetic$$$$progression,thenthecentreof$$$$incircleandthecentroidof$$$$trianglelieonalineparallelto$$$$thesideofmiddlelengthofthe$$$$triangle.$$

Answered by mr W last updated on 20/Jun/19

$$letthemiddlesidelengthbea,$$$$thentheothertwosidesare$$$$a-danda+d.$$$$theperimeterofthetriangleis$$$$p=a+(a-d)+(a+d)=3a.$$$$let\mathrm{\Delta }betheareaofthetriangle,$$$$theradiusoftheincircleisr,$$$$\frac{1}{2}pr=\mathrm{\Delta }$$$$\Rightarrow r=\frac{2\mathrm{\Delta }}{p}=\frac{2\mathrm{\Delta }}{3a}$$$$leth=altitudeovermiddleside$$$$\frac{1}{2}ah=\mathrm{\Delta }$$$$\Rightarrow h=\frac{2\mathrm{\Delta }}{a}$$$$thedistanceofthecentroidtothe$$$$middlesideis\frac{h}{3}=\frac{2\mathrm{\Delta }}{3a},whichis$$$$equaltotheradiusofincircle.that$$$$meansthecenterofincircleandthe$$$$centroidhavethesamedistanceto$$$$themiddleside,i.e.theylieona$$$$lineparalleltothemiddleside.$$

Commented by ajfour last updated on 20/Jun/19

$$ThankyouSir,verygoodapproach!$$

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