A circle with center
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
passes through the vertices
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
and intersects the segments
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
again at distinct points
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
and
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
respectively. Let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be the point of intersection of the circumcircles of triangles
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
and
![KBN](/media/m/9/3/e/93e05b5b31787d7c1bfe5228767ef764.png)
(apart from
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
). Prove that
![\angle OMB=90^{\circ}](/media/m/5/e/a/5ea68e012d839ad4bf9728dae83f9fc0.png)
.
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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 $\angle OMB=90^{\circ}$.