ILL 1983 Problem 47

In a plane, three pairwise intersecting circles $C_1, C_2, C_3$ with centers $M_1,M_2,M_3$ are given. For $i = 1, 2, 3$ , let $A_i$ be one of the points of intersection of $C_j$ and $C_k \ (\{i, j, k \} = \{1, 2, 3 \})$ . Prove that if $\angle M_3A_1M_2 = \angle M_1A_2M_3 = \angle M_2A_3M_1 = \frac{\pi}{3}$ (directed angles), then $M_1A_1, M_2A_2$ , and $M_3A_3$ are concurrent.