BALTIC-WAY 2007 Problem 14

In a convex quadrilateral $ABCD$ we have $ADC = 90^{\circ}$ . Let $E$ and $F$ be the projections of $B$ onto the lines $AD$ and $AC$ , respectively. Assume that $F$ lies between $A$ and $C$ , that $A$ lies between $D$ and $E$ , and that the line $EF$ passes through the midpoint of the segment $BD$ . Prove that the quadrilateral $ABCD$ is cyclic.