MEMO 2012 Problem 6

Let $ABCD$ be a convex quadrilateral with no pair of parallel sides, such that $\angle ABC = \angle CDA$ . Assume that the intersections of the pairs of neighbouring angle bisectors of $ABCD$ form a convex quadrilateral $EFGH$ . Let $K$ be the intersection of the diagonals of $EFGH$ . Prove that the lines $AB$ and $CD$ intersect on the circumcircle of the triangle $BKD$ .