IMO Shortlist 2009 problem G3
Dodao/la:
arhiva2. travnja 2012. Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle. The incircle of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
touches the sides
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
at the points
![Z](/media/m/7/9/4/794ff2bd637e30ea27e50e57eecd0b76.png)
and
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
, respectively. Let
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
be the point where the lines
![BY](/media/m/c/e/b/ceb91029af823458377562b596edba2f.png)
and
![CZ](/media/m/2/3/f/23fdaf71b1b1d944bf8a98d34bba554b.png)
meet, and let
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
and
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
be points such that the two quadrilaterals
![BCYR](/media/m/a/f/b/afb7d0829c9662b524b270b524cada80.png)
and
![BCSZ](/media/m/c/b/6/cb6d57ccb1fbc718e0f6df2208cc7dfc.png)
are parallelogram.
Prove that
![GR=GS](/media/m/b/5/a/b5afc3601665257890902813690cf113.png)
.
Proposed by Hossein Karke Abadi, Iran
%V0
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