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Beautiful geometry

Source: III Caucasus Mathematical Olympiad

March 17, 2018
geometry

Problem Statement

By centroid of a quadrilateral PQRSPQRS we call a common point of two lines through the midpoints of its opposite sides. Suppose that ABCDEFABCDEF is a hexagon inscribed into the circle Ω\Omega centered at OO. Let AB=DEAB=DE, and BC=EFBC=EF. Let XX, YY, and ZZ be centroids of ABDEABDE, BCEFBCEF; and CDFACDFA, respectively. Prove that OO is the orthocenter of triangle XYZXYZ.
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