BALTIC-WAY 1991 Problem 16

Let two circles $C_{1}$ and $C_{2}$ (with radii $r_{1}$ and $r_{2}$ ) touch each other externally, and let $l$ be their common tangent. A third circle $C_{3}$ (with radius $r_{3}<\min \left(r_{1}, r_{2}\right)$ ) is externally tangent to the two given circles and tangent to the line $l$. Prove that

$$\frac{1}{\sqrt{r_{3}}}=\frac{1}{\sqrt{r_{1}}}+\frac{1}{\sqrt{r_{2}}}$$