Given a segment A B in the plane, choose on it a point M different from A and B. Two equilateral triangles \triangle A M C and \triangle B M D in the plane are constructed on the same side of segment A B. The circumcircles of the two triangles intersect in point M and another point N. (The circumcircle of a triangle is the circle that passes through all three of its vertices.)
(a) Prove that lines A D and B C pass through point N.
(b) Prove that no matter where one chooses the point M along segment A B, all lines M N will pass through some fixed point K in the plane.