Midpoint Diagonal Quadrilaterals

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 $E_I$.