$A B$ and $A C$ are tangents to a circle with center $O, M$ is the intersection point of line $A O$ with the circle; $D E$ is a segment of the tangent line through point $M$, between $A B$ and $A C$. Find $D E$, if the radius of the circle is

15, and $A O=39$.
Consider similar triangles.

## Solution

$A M=A O-O M=24, A B^{2}=A O^{2}-O B^{2}=39^{2}-15^{2}=24 \cdot 54=36^{2}$. From the similarity of triangles $A M D$ and $A B O$, it follows that $D M={ }^{O B} / A B \cdot A M=10$. Therefore, $D E=2 D M=20$.

![](https://cdn.mathpix.com/cropped/2024_05_06_14311972f49418f31424g-16.jpg?height=286&width=504&top_left_y=646&top_left_x=777)

## Answer

20.