radius of a circle inscribed inside any triangle the radius of a circle tangent to 3 sides of a traiangle and inside it whose sides are $a,b,$ and $c$ is given by:$$r=\sqrt{\frac{(s-a)(s-b)(s-c)}{s}}$$
where $s=\frac{a+b+c}{2}$ , how this expression could be derived?

Hint: Note that $$A_\triangle=\frac{ar}{2}+\frac{br}{2}+\frac{cr}{2}=rS$$
and by Heron's formula:
$$A_\triangle=\sqrt{S(S-a)(S-b)(S-c)}$$
See; Heron's Formula
