$\left[\begin{array}{l}{[\text { Tangent Circles }]} \\ {[\text { Law of Cosines }}\end{array}\right]$

Two equal smaller circles are tangent to a larger circle - one internally and the other externally, and the arc between the points of tangency contains $60^{\circ}$. The radii of the smaller circles are $r$, and the radius of the larger circle is $R$. Find the distance between the centers of the smaller circles.
The line connecting the centers of two tangent circles passes through the point of tangency.

## Solution

Let $O, O_{1}$ and $O_{2}$ be the centers of the given circles. In triangle $O_{1} O_{2}$, it is known that

$$
O O_{1}=R-r, O O_{2}=R+r, \angle O_{1} O O_{2}=60^{\circ}
$$

By the cosine rule,

$$
\begin{aligned}
& O_{1} O_{2}^{2}=(R+r)^{2}+(R-r)^{2}-2(R+r)(R-r) \cos 60^{\circ}= \\
&=R^{2}-3 r^{2}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_1b17058527caf5b5d058g-08.jpg?height=380&width=372&top_left_y=2377&top_left_x=843)
Problem