OSN 2014 Problem 3

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$ . Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$ . Construct the parallelograms $AEDK$ and $BECL$ . Prove that $EF$ passes through the midpoint of the segment $KL$ .