\(\qquad\)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 the side \(BC\). The bisectors of the angles \(\angle BAC\) and \(\angle MON\) intersect at \(R\). \(\\\qquad\)Prove that the circumcircles of the triangles \(BMR\) and \(CNR\) have a common point lying on the side \(BC\).
| Attributes | Олімпіадна |
|---|---|
| Source | International Mathematical Olympiad |
| Year | 2004 |
| Number | 1 |
| Difficulty | 10.0 |
| Themes | Геометрія |