3 [ Auxiliary similar triangles

A circle of radius 1 touches a circle of radius 3 at point $C$. A line passing through point $C$ intersects the smaller circle at point $A$ and the larger circle at point $B$. Find $A C$, if $A B=2 \sqrt{5}$.

#
Connect the centers of the circles with the point of tangency and consider similar triangles.

## Solution

Let $O_{1}$ and $O_{2}$ be the centers of the smaller and larger circles, respectively. Since $\angle A C O_{1} = \angle B C O_{2}$ and triangles $A_{1} C$ and $B_{2} C$ are isosceles, these triangles are similar, and the similarity ratio is equal to the ratio of the radii of the circles, which is $1 / 3$.

If the circles touch internally, then $A C = 1 / 2 A B = \sqrt{5}$, which is impossible because the chord $A C$ of the smaller circle cannot be greater than the diameter of this circle.

Therefore, the circles touch externally, and $A C = 1 / 4 A B = \frac{\sqrt{5}}{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0ef0b80dacb6d98eff5ag-17.jpg?height=468&width=692&top_left_y=1804&top_left_x=683)

## Answer

$\frac{\sqrt{5}}{2}$