IMO 1992 LL MON50

Origin: MON

Problem

Let N be a point inside the triangle ABC. Through the mid- points of the segments AN, BN, and CN the lines parallel to the opposite sides of \triangleABC are constructed. Let AN, BN, and CN be the intersection points of these lines. If N is the orthocenter of the triangle ABC, prove that the nine-point circles of \triangleABC and \triangleANBNCN coincide. Remark. The statement of the original problem was that the nine-point circles of the triangles ANBNCN and AMBMCM coincide, where N and M are the orthocenter and the centroid of \triangleABC. This statement is false.