Hints

In any quadrilateral ABCD that
is not a parallelogram , if O' lies on MN prove that:
Area(AO'B) + Area(CO'D) = Area(AO'D) + Area(BO'C)

In a circumscribed quadrilateral
ABCD if O is the center of the circle inscribed prove that:
Area(AOB) + Area(COD) = Area(AOD) + Area(BOC)

From (1) and (2), O lies on MN.
See also:
Puzzle of
the Newton's Theorem: 50 pieces of circles. 