ILL 1985 Problem 52

In the triangle $ABC$ , let $B_1$ be on $AC, E$ on $AB, G$ on $BC$ , and let $EG$ be parallel to $AC$ . Furthermore, let $EG$ be tangent to the inscribed circle of the triangle $ABB_1$ and intersect $BB_1$ at $F$ . Let $r, r_1$ , and $r_2$ be the inradii of the triangles $ABC, ABB_1$ , and $BFG$ , respectively. Prove that $r = r_1 + r_2.$