Question 27
Consider $\triangle A B C$. Choose a point $M$ on its side $B C$ and let $O$ be the center of the circle passing through the vertices of $\triangle A B M$. Let $k$ be the circle that passes through $A$ and $M$ and whose center lies on line $B C$. Let line $M O$ intersect $k$ again in point $K$. Prove that the line $B K$ is the same for any choice of point $M$ on segment $B C$, so long as all of these constructions are well-defined.