Question

In a triangle ABC, $I$ be the incenter and $I_1,I_2$ and $I_3$ be the excenters. If R and r be the radius of circumcircle and incircle respectively, then $I{I}_1.I{I}_2.I{I}_3=$

• A. $\frac{R^{2}r}{16}$
• B. $4R^{2}r$
• C. $16R^{2}r$
• D. $16rR$

Answer 1

The correct option is C $16R^{2}r$

there's some standard result that says:

$I{I}_1=4Rsin\frac{A}{2}$

$I{I}_2=4Rsin\frac{B}{2}$

$I{I}_3=4Rsin\frac{C}{2}$

now $I{I}_1.I{I}_2.I{I}_3$ $=$ $\left(4R{)}^3\right[sin\frac{A}{2}.sin\frac{B}{2}.sin\frac{C}{2}]$ -----equ($1$)

we know that $r=4R.sin\frac{A}{2}.sin\frac{B}{2}.sin\frac{C}{2}$ ------equ($2$)

using equ($1$) and equ($2$)

we get $I{I}_1.I{I}_2.I{I}_3=16R^{2}r$