Mathematics - Properties of Triangle Question with Solution | TestHub

Let $\Delta$ be the area of the triangle and $s$ be its semi-perimeter. The formulas for the inradius ($r$) and exradii ($r_1,r_2,r_3$) are:$r=\frac{\Delta }{s}$$r_1=\frac{\Delta }{s-a}$$r_2=\frac{\Delta }{s-b}$$r_3=\frac{\Delta }{s-c}$ Now, let's calculate the product $r{r}_1{r}_2{r}_3$:$r{r}_1{r}_2{r}_3=\left(\frac{\Delta }{s}\right)\left(\frac{\Delta }{s-a}\right)\left(\frac{\Delta }{s-b}\right)\left(\frac{\Delta }{s-c}\right)=\frac{{\Delta }^4}{s(s-a)(s-b)(s-c)}$. According to Heron's formula for the area of a triangle, ${\Delta }^2=s(s-a)(s-b)(s-c)$. Substituting this into the expression for the product:$r{r}_1{r}_2{r}_3=\frac{{\Delta }^4}{{\Delta }^2}={\Delta }^2$.