Let `A B C` be a triangle with incenter `I` and inradius `rdot` Let `D ,E ,a n dF` be the feet of the perpendiculars from `I` to the sides `B C ,C A ,a n dA B ,` respectively. If `r_1,r_2a n dr_3` are the radii of circles inscribed in the quadrilaterals `A F I E ,B D I F ,a n dC E I D ,` respectively, prove that `(r_1)/(r-1_1)+(r_2)/(r-r_2)+(r_3)/(r-r_3)=(r_1r_2r_3)/((r-r_1)(r-r_2)(r-r_3))`

Explanation

In`/_AIF` and `/_AIE`

`(IF)/(sin(A/2))=AI=(IE)/(sin(A/2))`

`(AI)^2=(IE*IF)/(sin^2(A/2))`

`(BI)^2=(IF*ID)/(sin^2(B/2))`

`(CCI)^2=(IE*ID)/(sin(C/2))`

`(AI)^2*(BI)^2*(CI)^2=((IE)^2(IF)^2(ID)^2)/(sin^2(A/2)sin^2(B/2)sin^2(C/2))`

`(IE*IF*ID)/(AI*BI*CI)=sin(A/2)sin(B/2)sin(C/2)=r/(4R)`

`r=4Rsin(A/2)sin(B/2)sin(C/2)`

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