IMO 1970 Problem 1

$M$ is any point on the side $AB$ of the triangle $ABC$ . $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$ . $q$ is the radius of the circle on the opposite side of $AB$ to $C$ , touching the three sides of $AB$ and the extensions of $CA$ and $CB$ . Similarly, $q_1$ and $q_2$ . Prove that $r_1r_2q=rq_1q_2$ .