1. Let be an acute-angled triangle with . The circle with diameter intersects the sides and at and respectively. Denote by the midpoint of the side . The bisectors of the angles and intersect at . Prove that the circumcircles of the triangles and have a common point lying on the side .
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1. Let $ABC$ be an acute-angled triangle with $AB\neq AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of the side $BC$. The bisectors of the angles $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.
Izvor: Međunarodna matematička olimpijada, shortlist 2004
Školjka 
be an acute-angled triangle with 
. The circle with diameter 
intersects the sides 
and 
at 
and 
respectively. Denote by 
the midpoint of the side 
and 
intersect at 
. Prove that the circumcircles of the triangles 
and 
have a common point lying on the side