IMO Shortlist 2007 problem G1
Dodao/la:
arhiva2. travnja 2012. In triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
the bisector of angle
![BCA](/media/m/f/1/0/f10499c51292756d9ee2b050c9474965.png)
intersects the circumcircle again at
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
, the perpendicular bisector of
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
at
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
, and the perpendicular bisector of
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
at
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
. The midpoint of
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
is
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
and the midpoint of
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
is
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
. Prove that the triangles
![RPK](/media/m/2/b/a/2bac68b7f11a6d2be9a0e1f8dbc3b8ec.png)
and
![RQL](/media/m/8/4/3/84391367f98771fdaf07028e88a1eaad.png)
have the same area.
Author: Marek Pechal, Czech Republic
%V0
In triangle $ABC$ the bisector of angle $BCA$ intersects the circumcircle again at $R$, the perpendicular bisector of $BC$ at $P$, and the perpendicular bisector of $AC$ at $Q$. The midpoint of $BC$ is $K$ and the midpoint of $AC$ is $L$. Prove that the triangles $RPK$ and $RQL$ have the same area.
Author: Marek Pechal, Czech Republic
Izvor: Međunarodna matematička olimpijada, shortlist 2007