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Question

In a triangle ABC, I be the incenter and I1,I2 and I3 be the excenters. If R and r be the radius of circumcircle and incircle respectively, then II1.II2.II3=

A
R2r16
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B
4R2r
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C
16R2r
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D
16rR
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Solution

The correct option is C 16R2r
there's some standard result that says:
II1=4RsinA2

II2=4RsinB2

II3=4RsinC2

now II1.II2.II3 = (4R)3[sinA2.sinB2.sinC2] -----equ(1)

we know that r=4R.sinA2.sinB2.sinC2 ------equ(2)

using equ(1) and equ(2)
we get II1.II2.II3=16R2r

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