KöMaL 2026 B5533

In acute triangle $ABC$ let $A_0$ be the foot of the altitude from $A$, and let $B_0$ be the foot of the altitude from $B$. On circle $A_0B_0C$ let $X$ and $Y$ be the two points for which circles $ABX$ and $ABY$ are tangent to circle $A_0B_0C$. Prove that line $XY$ bisects side $AB$.