Ah yes, I see it IS all quadrilaterals (for which the segments joining the midpoints of pairs of opposite sides meet) - for the midpoint of (A+B)/2 and (C+D)/2 is (A+B+C+D)/4, the centroid of the four vertices. So indeed in ANY quadrilateral these "bisector segments" bisect each other. Nice try, though.
I don't regard the fact that the bisector segment from AB to CD bisects the area as really different from the fact that the area formula works for trapezoids. You're going to have a hard time convincing me that trapezoids were really worth naming!
I wonder - if one bisector segment of a quadrilateral DOES bisect it, is it necessarily a trapezoid? It feels as though it might be.