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 $BC$.
Školjka