Let be a triangle. The incircle of touches the sides and at the points and , respectively. Let be the point where the lines and meet, and let and be points such that the two quadrilaterals and are parallelogram. Prove that .
Proposed by Hossein Karke Abadi, Iran
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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
Izvor: Međunarodna matematička olimpijada, shortlist 2009
Školjka 
be a triangle. The incircle of 
and 
at the points 
and 
, respectively. Let 
be the point where the lines 
and 
meet, and let 
and 
be points such that the two quadrilaterals 
and 
are parallelogram.
.