Question 2
Let $ABCD$ be a convex quadrilateral. A point $M$ moves on line $AB$, with $M\ne A,B$. Let $N$ be the second intersection, distinct from $M$, of the circles $(MAC)$ and $(MBD)$. Prove that:
(1) $N$ lies on a fixed circle;
(2) line $MN$ always passes through a fixed point.
Here $(XYZ)$ denotes the circle through $X,Y,Z$.
(1) $N$ lies on a fixed circle;
(2) line $MN$ always passes through a fixed point.
Here $(XYZ)$ denotes the circle through $X,Y,Z$.