International Mathematical Olympiad 1970 Problem 1

Let $M$ be a point on the side $AB$ of $\triangle ABC$. Let $r_1, r_2$ and $r$ be the radii of the inscribed circles of triangles $AMC, BMC$ and $ABC$. Let $q_1, q_2$ and $q$ be the radii of the escribed circles of the same triangles that lie in the angle $ACB$. Prove that $$\frac{r_1}{q_1} \cdot \frac{r_2}{q_2} = \frac{r}{q}.$$