VMO 2006 Problem 2

Let $ABCD$ be a convex quadrilateral. Take an arbitrary point $M$ on the line $AB$ , and let $N$ be the point of intersection of the circumcircles of triangles $MAC$ and $MBC$ (different from $M$ ). Prove that:

a) The point $N$ lies on a fixed circle;

b) The line $MN$ passes though a fixed point.