A circle with center
passes through the vertices
and
of the triangle
and intersects the segments
and
again at distinct points
and
respectively. Let
be the point of intersection of the circumcircles of triangles
and
(apart from
). Prove that
.
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A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.
Školjka