Question 114
In triangle $ABC$, arbitrary points $P, Q$ lie on side $BC$ such that $BP = CQ$ and $P$ lies between $B, Q$. The circumcircle of triangle $APQ$ intersects sides $AB$ and $AC$ at $E$ and $F$, respectively. The point $T$ is the intersection point of $EP$ and $FQ$. Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X$ and $Y$, respectively. Prove that the circumcircles of triangles $TXY$ and $APQ$ are tangent to each other.