Question 133
Let $ABCD$ be a convex quadrangle such that $\angle ABC = \angle ADC \lt 90^\circ$. The internal bisectrices of the angles $ABC$ and $ADC$ cross the diagonal $AC$ at $E$ and $F$, respectively, and meet at $P$. Let $M$ be the midpoint of the diagonal $AC$. The segments $BM$ and $DM$ cross the circle $BDP$ again at $X$ and $Y$, respectively, and the lines $EX$ and $FY$ meet at $Q$. Prove that the lines $AC$ and $PQ$ are perpendicular.
IMO 2016 Shortlist
IMO 2016 Shortlist