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d does not depend on the order of the axes of symmetry

Source: Vietnam TST 1996 for the 37th IMO, problem 4

June 26, 2005
symmetrygeometry unsolvedgeometry

Problem Statement

Given 3 non-collinear points A,B,CA,B,C. For each point MM in the plane (ABCABC) let M1M_1 be the point symmetric to MM with respect to ABAB, M2M_2 be the point symmetric to M1M_1 with respect to BCBC and M′M' be the point symmetric to M2M_2 with respect to ACAC. Find all points MM such that MM′MM' obtains its minimum. Let this minimum value be dd. Prove that dd does not depend on the order of the axes of symmetry we chose (we have 3 available axes, that is BCBC, CACA, ABAB. In the first part the order of axes we chose ABAB, BCBC, CACA, and the second part of the problem states that the value dd doesn't depend on this order).
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