SHARYGIN 2008 Problem 9

(A.Zaslavsky, 9--10) The reflections of diagonal $BD$ of a quadrilateral $ABCD$ in the bisectors of angles $B$ and $D$ pass through the midpoint of diagonal $AC$ . Prove that the reflections of diagonal $AC$ in the bisectors of angles $A$ and $C$ pass through the midpoint of diagonal $BD$ (There was an error in published condition of this problem).