Let $ABC$ be a triangle and let $D, E$ are points on its circumscribed circle, such that $D$ lies on arc $AB, E$ lies on arc $AC$ (smaller arcs) and $BD \parallel CE$ . Let the point F be the intersection of the lines $DA$ and $CE$, and the intersection of the lines $EA$ and $BD$ is $G$. Let $P$ be the second intersection of the circumscribed circles of $\vartriangle ABG$ and $\vartriangle ACF$. Prove that the line$ AP$ passes through the mid point of the side $BC$.