KöMaL 2025 B5497

Circular arc $k$ with endpoints $AB$ is given. Let $F$ be the midpoint of the arc. A circle is tangent internally to $k$ at point $P$, and tangent internally to line segment $AB$ at point $Q$. Prove that the sum of the radii of circle $APQ$ and $BPQ$ equals the length of line segment $AF$.