IMO Shortlist 2015 problem G5

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Let ABC be a triangle with CA \neq CB. Let D, F, and G be the midpoints of the sides AB, AC, and BC respectively. A circle \Gamma passing through C and tangent to AB at D meets the segments AF and BG at H and I, respectively. The points H' and I' are symmetric to H and I about F and G, respectively. The line H'I' meets CD and FG at Q and M, respectively. The line CM meets \Gamma again at P. Prove that CQ = QP.

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