BY-MO 2004 Problem 2

Let $C$ be a semicircle with diameter $AB$ . Circles $S$ , $S_1$ , $S_2$ with radii $r$ , $r_1$ , $r_2$ , respectively, are tangent to $C$ and the segment $AB$ , and moreover $S_1$ and $S_2$ are externally tangent to $S$ . Prove that $\frac{1}{\sqrt{r_1}}+\frac{1}{\sqrt{r_2}}=\frac{2\sqrt{2}}{\sqrt{r}}$