Each of the three circles of radius $r$ touches the other two. Find the area of the triangle formed by the common external tangents to these circles.
The given triangle is equilateral.

## Solution

The given triangle is equilateral because its sides are parallel to the sides of the triangle with vertices at the centers of the given circles. The side of the given triangle is

$$
r + r + r \sqrt{3} + r \sqrt{3} = 2 r (\sqrt{3} + 1)
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_4fce0705b02fa9e106d4g-07.jpg?height=443&width=514&top_left_y=2237&top_left_x=772)

## Answer

$2 r^{2}(2 \sqrt{3} + 3)$.

Two equal circles with centers $O_{1}$ and $O_{2}$ touch each other internally and touch a circle of radius $R$ with center $O$.

Find the perimeter of triangle $O O_{1} O_{2}$.

## Hint

The line connecting the centers of two touching circles passes through their point of tangency.

## Solution

Let $r$ be the radius of the smaller circles. Then $O O_{1} = O O_{2} = R - r$, and $O_{1} O_{2} = 2 r$. Therefore, the desired perimeter is $2(R - r) + 2 r = 2 R$.

![](https://cdn.mathpix.com/cropped/2024_05_06_4fce0705b02fa9e106d4g-08.jpg?height=395&width=418&top_left_y=757&top_left_x=823)

## Answer

$2 R$.