  A convex quadrilateral, $Q$, is called a midpoint diagonal quadrilateral if
the intersection point of the diagonals of $Q$ coincides with the midpoint of
at least one of the diagonals of $Q$. A parallelogram, P, is a special case of
a midpoint diagonal quadrilateral since the diagonals of P bisect one another.
We prove two results about ellipses inscribed in midpoint diagonal
quadrilaterals, which generalize properties of ellipses inscribed in
parallelograms involving convex quadrilaterals. First, $Q$ is a midpoint
diagonal quadrilateral if and only if each ellipse inscribed in $Q$ has
tangency chords which are parallel to one of the diagonals of $Q$. Second, $Q$
is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in
$Q$ has a unique pair of conjugate diameters parallel to the diagonals of $Q$.
Finally, we show that there is a unique ellipse, $E_I$, of minimal eccentricity
inscribed in a midpoint diagonal quadrilateral, $Q$, and also that the unique
pair of conjugate diameters parallel to the diagonals of $Q$ are the equal
conjugate diameters of $E_I$.
