ILL 1982 Problem 54

The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$ , and we also have $\angle CAB = \angle C_1AB_1$ . Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$ . Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.