Question

lf $r_1,r_2,r_3$ are the radii of the escribed circles of a $\Delta ABC$ and if $r$ is the radius of its incircle then $r_1{r}_2{r}_3-r(r_1{r}_2+r_2{r}_3+r_3{r}_1)=$

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is A 0

$r_1=\frac{\Delta }{s-a},r_2=\frac{\Delta }{s-b},r_3=\frac{\Delta }{s-c},r=\frac{\Delta }{s}$

now substitute these values in the expression in question we get

$\frac{{\Delta }^3}{(s-a)(s-b)(s-c)}-\frac{{\Delta }^3}{s}(\frac{1}{(s-a)(s-b)}+\frac{1}{(s-b)(s-c)}+\frac{1}{(s-c)(s-a)})$

$\frac{{\Delta }^3}{s(s-a)(s-b)(s-c)}(s-c(s-a\left)\right(s-b\left)\right(s-c\left)\right)$

$\frac{{\Delta }^3(s-s)}{s(s-a)(s-b)(s-c)}$

$=0$.