AUSTRIAN-POLISH 2005 Problem 5

Given is a convex quadrilateral $ABCD$ with $AB=CD$ . Draw the triangles $ABE$ and $CDF$ outside $ABCD$ so that $\angle{ABE} = \angle{DCF}$ and $\angle{BAE}=\angle{FDC}$ . Prove that the midpoints of $\overline{AD}$ , $\overline{BC}$ and $\overline{EF}$ are collinear.