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equilateral wanted if 3 projections are equal

Source: VI Soros Olympiad 1990-00 R1 9.7 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 27, 2024
geometryEquilateral

Problem Statement

In the acute-angled triangle ABCABC, the points PP, NN, M M are the feet of the altitudes drawn from the vertices CC, AA, BB, respectively. The lengths of the projections of the sides ABAB, BCBC, CACA on straight lines MNMN, PMPM, NPNP respectively, are equal to each other. Prove that triangle ABCABC is regular.
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