Question 8
(POL 2; IMO 1) Given a point $M$ on the side $AB$ of the triangle $ABC$, let $r_1$ and $r_2$ be the radii of the inscribed circles of the triangles $ACM$ and $BCM$ respectively and let $\rho_1$ and $\rho_2$ be the radii of the excircles of the triangles $ACM$ and $BCM$ at the sides $AM$ and $BM$ respectively. Let $r$ and $\rho$ denote the radii of the inscribed circle and the excircle at the side $AB$ of the triangle $ABC$ respectively. Prove that
$$\frac{r_1r_2}{\rho_1\rho_2}=\frac r\rho.$$
$$\frac{r_1r_2}{\rho_1\rho_2}=\frac r\rho.$$