The so-called Butterfly Theorem says:
If are two chords of a circle and , and is another chord of that circle which passes through , and , , then we have if and only if .
Back to the problem: Let intersect the circumcircle of again in , and let . Since , the condition of the theorem is satisfied (if intersects the circumcircle in , then , hence we also have ).
But it is well known that and are symmetric wrt the line , thus also . This implies that is a parallelogramm, and consequently