For triangle `ABC,R=5/2 and r=1.` Let `I` be the incenter of the triangle and `D,E and F` be the feet of the perpendiculars from `I->BC,CA and AB,` respectively. The value of `(ID*IE*IF)/(IA*IB*IC)` is equal to (a) `5/2` (b) `5/4` (c) `1/10` (d) `1/5`

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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