IMO Shortlist 2009 problem G3

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Let ABC be a triangle. The incircle of ABC touches the sides AB and AC at the points Z and Y, respectively. Let G be the point where the lines BY and CZ meet, and let R and S be points such that the two quadrilaterals BCYR and BCSZ are parallelogram.
Prove that GR=GS.

Proposed by Hossein Karke Abadi, Iran
Source: Međunarodna matematička olimpijada, shortlist 2009