Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively. Let $r_1{r}_2$ and $r_3$ be the ex-radii of the triangle.

Then $r_1{r}_2+r_2{r}_3+r_3{r}_1=?$

• A. $\frac{{\Delta }^2}{r^2}$
• B. $\frac{\Delta }{r}$
• C. $\frac{2\Delta }{r}$
• D. ${\Delta }^2$

Answer 1

The correct option is A $\frac{{\Delta }^2}{r^2}$

Sides of $\Delta$ $ABC$ are denoted by $a,b,c$ respectively.

If $r_1{r}_2$ and $r_3$ be the ex-radii of the triangle.

Then

$r_1=\frac{\Delta }{s-a}$

$r_2=\frac{\Delta }{s-b}$

$r_3=\frac{\Delta }{s-c}$

$\therefore r_1{r}_2+r_2{r}_3+r_3{r}_1$

$=\frac{{\Delta }^2}{(s-a)(s-b)}$$+\frac{{\Delta }^2}{(s-b)(s-c)}$$+\frac{{\Delta }^2}{(s-c)(s-a)}$

$=(\frac{{\Delta }^2(s-c+s-a+s-b)}{(s-a)(s-b)(s-c)})$

We know $2s=a+b+c$ and ${\Delta }^2=s(s-a)(s-b)(s-c)$

$\therefore r_1{r}_2+r_2{r}_3+r_3{r}_1=s^2$

$s^2=\frac{{\Delta }^2}{r^2}$ $(\because \Delta =rs)$