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perpendiculars from P on equilateral ABC, centroid of projections', midpoint

Source: Sharygin 2005 IX CR 9

August 18, 2019
EquilateralEquilateral TriangleperpendicularmidpointCentroidgeometry

Problem Statement

Let OO be the center of a regular triangle ABCABC. From an arbitrary point PP of the plane, the perpendiculars were drawn on the sides of the triangle. Let MM denote the intersection point of the medians of the triangle , having vertices the feet of the perpendiculars. Prove that MM is the midpoint of the segment POPO.
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