Chapter 8.4, Problem 58AE

Inscribed Circle For Problems 55-58, the lines that bisect each angle of a triangle meet in a single point $O$, and the perpendicular distance $r$ from $O$ to each side of the triangle is the same. The circle with center at $O$ and radius $r$ is called the inscribed circle of the triangle (see the figure).

Show that the area $K$ of triangle $PQR$ is $K=rs$, where $s=\frac{1}{2}\left(a+b+c\right)$.

Then show that $r=\sqrt{\frac{\left(s-a\right)\left(s-b\right)\left(s-c\right)}{s}}$