CZ-SK-MO 1986 Problem 4

Let $C_1,C_2$ , and $C_3$ be points inside a bounded convex planar set $M$ . Rays $l_1,l_2,l_3$ emanating from $C_1,C_2,C_3$ respectively partition the complement of the set $M \cup l_1 \cup l_2 \cup l_3$ into three regions $D_1,D_2,D_3$ . Prove that if the convex sets $A$ and $B$ satisfy $A\cap l_j =\emptyset = B\cap l_j$ and $A\cap D_j \ne \emptyset \ne B\cap D_j$ for $j = 1,2,3$ , then $A\cap B \ne \emptyset$