Two equal circles touch the inside of a third circle and touch each other. By connecting the three centers, a triangle with a perimeter of 18 is obtained. Find the radius of the larger circle.
The line connecting the centers of two tangent circles passes through their point of tangency.

## Solution

Let the radii of the given circles be $r, r$ and $R (r < R)$. Then the sides of the specified triangle are $R - r, R - r$, and $2r$. Therefore,

Consequently, $R = 9$.

![](https://cdn.mathpix.com/cropped/2024_05_06_67482594d9a3488694ecg-23.jpg?height=406&width=778&top_left_y=100&top_left_x=640)

Answer

9.

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