Given an isosceles triangle $A B C$ with base $A C$. A circle of radius $R$ with center at point $O$ passes through points $A$ and $B$ and intersects the line $B C$ at point $M$, distinct from $B$ and $C$. Find the distance from point $O$ to the center of the circumcircle of triangle $A C M$.
Let $O_{1}$ be the center of the second circle. Then $\angle A O_{1} M=2 \angle C=180^{\circ}-\angle B$. Therefore, points $A, B, M$ and $O_{1}$ lie on the same circle. Hence, point $O_{1}$ belongs to the circumcircle of triangle $A B M$, which means $O O_{1}=R$.

![](https://cdn.mathpix.com/cropped/2024_05_06_df2bbd76ac4dd2f0f20fg-07.jpg?height=697&width=926&top_left_y=0&top_left_x=569)

## Answer

$R$.