Let ABCD be a convex quadrilateral
and let E and F be the points of
intersections of the lines AB, CD and
AD, BC, respectively. Prove that the
midpoints of the segments AC, BD,
and EF are collinear.
Let ABCD be a convex quadrilateral
and M a point in its interior such that
[MAB] = [MBC] = [MCD] = [MDA].
Prove that one of the diagonals of
ABCD passes through the midpoint of
the other diagonal.