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DM // AO, circumcircle, midpoint projections on sides, circumcenter

Source: 2018 Romania JBMO TST 4.2

June 1, 2020
geometrycircumcirclemidpointparallel

Problem Statement

Let ABCABC be an acute triangle, with AB≠ACAB \ne AC. Let DD be the midpoint of the line segment BCBC, and let EE and FF be the projections of DD onto the sides ABAB and ACAC, respectively. If MM is the midpoint of the line segment EFEF, and OO is the circumcenter of triangle ABCABC, prove that the lines DMDM and AOAO are parallel.
As source was given [url=https://artofproblemsolving.com/community/c629086_caucasus_mathematical_olympiad]Caucasus MO, but I was unable to find this problem in the contest collections
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