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starting with 2 points on sides of an angle, midpoints given and wanted

Source: 2007 Sharygin Geometry Olympiad Correspondence Round P16

April 27, 2019
geometryanglemidpoint

Problem Statement

On two sides of an angle, points A,BA, B are chosen. The midpoint MM of the segment ABAB belongs to two lines such that one of them meets the sides of the angle at points A1,B1A_1, B_1, and the other at points A2,B2A_2, B_2. The lines A1B2A_1B_2 and A2B1A_2B_1 meet ABAB at points PP and QQ. Prove that MM is the midpoint of PQPQ.
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