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.