RO-TST 1996 Problem 4

Let $ABCD$ be a cyclic quadrilateral and let $M$ be the set of incenters and excenters of the triangles $BCD$ , $CDA$ , $DAB$ , $ABC$ (so 16 points in total). Prove that there exist two sets $\mathcal{K}$ and $\mathcal{L}$ of four parallel lines each, such that every line in $\mathcal{K} \cup \mathcal{L}$ contains exactly four points of $M$ .