1. Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be an acute-angled triangle with
![AB\neq AC](/media/m/4/e/6/4e6e8246f2c52b45bd4d4c468a6d5fc2.png)
. The circle with diameter
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
intersects the sides
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
at
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
and
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
respectively. Denote by
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
the midpoint of the side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
. The bisectors of the angles
![\angle BAC](/media/m/b/2/1/b21a9e466104c5d33646432221e142be.png)
and
![\angle MON](/media/m/0/b/1/0b1565cb2a8561bf19d6b13b505fe834.png)
intersect at
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
. Prove that the circumcircles of the triangles
![BMR](/media/m/e/1/9/e19cbc2f10621872f260d68d06c06759.png)
and
![CNR](/media/m/c/6/8/c683d46c6b8936ec17a4ec04ada76c77.png)
have a common point lying on the side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
.
%V0
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 $BC$.