KöMaL 2025 B5437

Let $ABC$ be a triangle satisfying $BC<AC$, and let $D$ be a point on side $AC$ satisfying $BC=DC$. Let circle $\Gamma$ be tangent to arc $AC$ not containing $B$ of the circumcircle of $ABC$ at point $X$, and to line segment $AC$ at $D$. Let the tangent line from $B$ to $\Gamma$ that is closer to $C$ intersect $AC$ at $Y$, and be tangent to $\Gamma$ at $Z$. Prove that if $A$, $Y$, $X$ and the circumcenter of triangle $ABC$ are concyclic, then $AZ=CZ$.