ILL 1990 Problem 81

A circle of radius $\rho$ is tangent to the sides $AB$ and $AC$ of the triangle $ABC$, and its center $K$ is at a distance $p$ from $BC$.

(a) Prove that $a(p - \rho) = 2s(r - \rho)$, where $r$ is the inradius and $2s$ the perimeter of $ABC$. (b) Prove that if the circle intersect $BC$ at $D$ and $E$, then $$DE=\frac{4\sqrt{rr_1(\rho-r)(r_1-\rho)}}{r_1-r}$$

where $r_1$ is the exradius corresponding to the vertex $A.$