69th Belarusian Mathematical Olympiad

A point $P$ is chosen in the interior of the side $BC$ of the triangle $ABC$. The points $D$ and $E$ are symmetric to $P$ with respect to the vertices $B$ and $C$ respectively. The circumcircles of the triangles $ABE$ and $ACD$ intersect at the points $A$ and $X$. The ray $AB$ intersects the segment $XD$ at the point $C_1$ and the ray $AC$ intersects the segment $XE$ at the point $B_1$. Prove that the lines $BC$ and $B_1{C}_1$ are parallel.