BG-MO 2002 Problem 2

Consider the orthogonal projections of the vertices $A$ , $B$ and $C$ of triangle $ABC$ on external bisectors of $\angle ACB$ , $\angle BAC$ and $\angle ABC$ , respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$ , prove that $r^2+p^2=d^2$ Proposed by Alexander Ivanov