Given a triangle ABC. A point X is chosen on a side AC. Some circle passes through X, touches the side AC and intersects the circumcircle of triangle ABC in points M and N such that the segment MN bisects BX and intersects sides AB and BC in points P and Q. Prove that the circumcircle of triangle PBQ passes through a fixed point different from B.
proposed by Sergej Berlov geometrycircumcircleparallelogramconicsparabolaratiogeometric transformation