Let $P$ be the midpoint of the arc $AC$ of a circle, and $B$ be a point on the arc $AP$. Let $M$ and $N$ be the projections of $P$ onto the segments $AC$ and $BC$ respectively. Prove that if $D$ is the intersection of the bisector of $\angle ABC$ and the segment $AC$, then every diagonal of the quadrilateral $BDMN$ bisects the area of the triangle $ABC$.