Τρίτη, 17 Οκτωβρίου 2017

2004 IMO Shortlist G1

Let $ABC$ be an acute-angled triangle with $AB \ne AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ , respectively. Denote by $O$ the midpoint of $BC$. The bisectors of the angles $BAC$ and $MON$ intersect at $R$. Prove  that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the line segment $BC$.

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