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segments bisecting areas of trapezoid, symmetric intersection points

Source: - All-Russian MO 2002 Regional (R4) 9.6

September 17, 2024
geometrytrapezoidareassymmetry

Problem Statement

Let A′A' be a point on one of the sides of the trapezoid ABCDABCD such that line AA′AA' divides the area of the trapezoid in half. Points B′B', C′C', D′D' are defined similarly. Prove that the intersection points of the diagonals of quadrilaterals ABCDABCD and A′B′C′D′A'B'C'D' are symmetrical wrt the midpoint of midline of trapezoid ABCDABCD.
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