Léon Anne's Theorem

Pick a point $O$ in the interior of a Quadrilateral which is not a Parallelogram. Join this point to each of the four Vertices, then the Locus of points $O$ for which the sum of opposite Triangle areas is half the Quadrilateral Area is the line joining the Midpoints $M_1$ and $M_2$ of the Diagonals.

See also Diagonal (Polygon), Midpoint, Quadrilateral

References

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 174-175, 1991.