# Problem 4. (3 points)

Circles $O_{1}$ and $O_{2}$ touch circle $O_{3}$ with radius 13 at points $A$ and $B$ respectively and pass through its center $O$. These circles intersect again at point $C$. It is known that $O C=12$. Find $A B$.
# Answer: 10

## Solution:

Since circles $O_{1}$ and $O_{2}$ touch circle $O_{3}$ at points $A$ and $B$ respectively and pass through its center $O$, $AO$ and $BO$ are their diameters. Therefore, angles $\angle OCA$ and $\angle OCB$ are right angles. These cannot be the same angle, as this would mean that circles $O_{1}$ and $O_{2}$ coincide, so they are adjacent angles. Right triangles $OCA$ and $OCB$ are equal by the leg $OC$ and hypotenuses. Therefore, $AB = 2AC = 2\sqrt{OA^2 - OC^2} = 2\sqrt{13^2 - 12^2} = 10$.