Always, Sometimes, Never

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A circle with center $O$ passes through the vertices $A$ and $C$ of a non-degenerate triangle $ABC$ and intersects segments $AB$ and $BC$ again at distinct points $K$ and $N$ , respectively. The circumcircles of triangles ABC and KBN intersects at exactly two distinct points B and M. Is the statement true?

$\angle BMO = 90^{\circ}.$

Always Sometimes Never

1 solution

Alan Yan
Nov 14, 2015

It is well-known that $M$ is the Miquel Point of the quadrilateral $KNCA$ . Denote $M_1, M_2$ by the midpoints of $AK, NC$ respectively. It is well-known that we have a spiral similarity centered at $M$ that maps $KA \rightarrow NC$ which implies that $M_1 \rightarrow M_2$ . Then, it is well-known that we have that $B, M_1, M_2, M$ , due to $\text{Yufei Zhao's fact 5}$ ,are concylic. However, we have $M_1, O, M_2, B$ are concyclic. Therefore, this implies that $O, M_1, M_2, B, M$ are concyclic with $OB$ diameter. Thus, $\angle OMB = 90^{\circ}$ . So the answer is always.

Always, Sometimes, Never

1 solution

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