Given two concentric circles with radii 1 and 3 and a common center $O$. A third circle is tangent to both of them. Find the angle between the tangents to the third circle drawn from the point $O$.

#
The line of centers of two tangent circles passes through the point of tangency.

## Solution

Let $O_{1}$ be the center of the third circle, $O A$ and $O B$ be the tangents to it (A and $B$ are the points of tangency). Then $O O_{1}$ is the bisector of angle $A O B$,

$$
A O_{1}=1, O O_{1}=2, \angle O A O_{1}=90^{\circ}
$$

Therefore, $\angle A O O_{1}=30^{\circ}$, and $\angle A O B=60^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_c271881b12104a3b803eg-37.jpg?height=355&width=368&top_left_y=637&top_left_x=845)

## Answer

$60^{\circ}$.