IMO 1982 SL 20

C8 (TUN 3) Let ABCD be a convex quadrilateral and draw regular tri-

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IMO 1982 SL 20

Problem

C8 (TUN 3) Let ABCD be a convex quadrilateral and draw regular tri- angles ABM, CDP, BCN, ADQ, the first two outward and the other two inward. Prove that MN = AC. What can be said about the quadrilateral MNPQ?

Solution

Since MN is the image of AC under rotation about B for 60◦, we have MN = AC. Similarly, PQ is the image of AC under rotation about D through 60◦, from which it follows that PQ \parallelMN. Hence either M, N, P, Q are collinear or MNPQ is a parallelogram.