Let $ABCD$ be a convex quadrilateral inscribed in a circle with $M$ and $N$ the midpoints of the diagonals $AC$ and $BD$ respectively. Suppose that $AC$ bisects $\angle BMD$. Prove that $BD$ bisects $\angle ANC$.
Source: Singapore IMO TST 2007, Problem 2
Tags: geometry proposed, geometry
Let $ABCD$ be a convex quadrilateral inscribed in a circle with $M$ and $N$ the midpoints of the diagonals $AC$ and $BD$ respectively. Suppose that $AC$ bisects $\angle BMD$. Prove that $BD$ bisects $\angle ANC$.