[ Tangent Circles]  Similarity Criteria $\quad]$

Two circles touch each other externally. A line passing through the point of tangency forms chords in the circles, one of which is $13 / 5$ times the other. Find the radii of the circles if the distance between the centers is 36.
Connect the centers of the circles with the corresponding ends of the chords and consider similar triangles.

## Solution

Let $O_{1}$ and $O_{2}$ be the centers of the circles, $K$ be the point of tangency, $A K$ and $K B$ be the chords, $A K=5 / 13 B K$. $A O_{1} K$ and $B O_{2} K$ are isosceles triangles with equal angles at the base. Therefore, they are similar with a similarity ratio of ${ }^{A K} /{ }_{K B}=5 / 13$. Consequently, $K O_{1}=5 / 18 O_{1} O_{2}=10$, $K O_{2}=36-10=26$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0ef0b80dacb6d98eff5ag-12.jpg?height=326&width=489&top_left_y=2441&top_left_x=790)

## Problem