KöMaL 2024 C1836

Circle $k$ containing the endpoints of the base $AB$ of isosceles triangle $ABC$ touches lines $AC$ and $BC$ at $A$ and $B$. Let $P$ be a point of circle $k$ that is different from points $A$ and $B$. Prove that the distance of $P$ from side $AB$ is at most the average of its distances from sides $AC$ and $BC$.