[ Inscribed Circle in an Angle ] [ Two Pairs of Similar Triangles

Three circles are inscribed in an angle - a small, a medium, and a large one. The large circle passes through the center of the medium one, and the medium circle passes through the center of the small one.

Calculate the radii of the medium and large circles if the radius of the small circle is $r$ and the distance from its center to the vertex of the angle is $a$.

#
Consider similar right triangles.

## Solution

Let $R$ be the radius of the middle circle. Draw radii of the small and middle circles to the points of tangency with one of the sides of the given angle. From the similarity of the obtained right triangles, it follows that $r^{\prime} / a = R / (R + a)$.

From this, $R = a r / (a - r)$.

Knowing the radius of the middle circle, we can similarly find the radius of the large circle.

![](https://cdn.mathpix.com/cropped/2024_05_06_36c4956010bd91c4b4eeg-15.jpg?height=309&width=764&top_left_y=97&top_left_x=653)

## Answer

$\frac{a r}{a-r}, \frac{a r^{2}}{(a-r)^{2}}$