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P_1P_2,P_2P_3,...,P_nP_1 have equal lengths and midpoints A_1, A_2, A_3, A_1,..

Source: Austrian Polish 1991 APMC

April 30, 2020
equal segmentsmidpointscombinatorial geometrycombinatoricsgeometry

Problem Statement

Given two distinct points A1,A2A_1,A_2 in the plane, determine all possible positions of a point A3A_3 with the following property: There exists an array of (not necessarily distinct) points P1,P2,...,PnP_1,P_2,...,P_n for some n≥3n \ge 3 such that the segments P1P2,P2P3,...,PnP1P_1P_2,P_2P_3,...,P_nP_1 have equal lengths and their midpoints are A1,A2,A3,A1,A2,A3,...A_1, A_2, A_3, A_1, A_2, A_3, ... in this order.
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