In the triangle $ABC$, we have that $M$ and $N$ are points on $AB$ and $AC$, respectively, such that $BC$ is parallel to $MN$. A point $D$ is chosen inside the triangle $AMN$. Let $E$ and $F$ be the points of intersection of $MN$ with $BD$ and $CD$, respectively. Show that the line joining the centers of the circles circumscribed to the triangles $DEN$ and $DFM$ is perpendicular to $AD$.