Two circles of radius $r$ touch each other. In addition, each of them touches externally a third circle of radius $R$ at points $A$ and $B$ respectively.

Find the radius $r$, if $A B=12, R=8$.
Let $O_{1}$ and $O_{2}$ be the centers of circles with radius $r$, and $O$ be the center of a circle with radius $R$. Then triangles $O A B$ and $O O_{1} O_{2}$ are similar. Therefore, $O A: O O_{1}=A B: O_{1} O_{2}$, or $8 / 8+r=12 / 2 r$. From this, $r=24$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0ef0b80dacb6d98eff5ag-09.jpg?height=385&width=520&top_left_y=2037&top_left_x=769)

## Answer