  In an earlier paper of the author, we showed that there is a unique ellipse
of minimal eccentricity, $E_I$, inscribed in any convex quadrilateral, $Q$.
Using a different approach in this paper, we prove that there is a unique
ellipse of minimal eccentricity, $E_I$, inscribed in a midpoint diagonal
quadrilateral, $Q$, which is a quadrilateral with the property that the
intersection point of the diagonals of $Q$ coincides with the midpoint of at
least one of the diagonals of $Q$. Our main result is that if $Q$ is a midpoint
diagonal quadrilateral, then the smallest non-negative angle between equal
conjugate diameters of $E_I$ equals the smallest non-negative angle between the
diagonals of $Q$. This was proven in another earlier paper of the author for
parallelograms.
