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In an earlier paper of the author, we showed that there is a unique ellipseof minimal eccentricity, $E_I$, inscribed in any convex quadrilateral, $Q$.Using a different approach in this paper, we prove that there is a uniqueellipse of minimal eccentricity, $E_I$, inscribed in a midpoint diagonalquadrilateral, $Q$, which is a quadrilateral with the property that theintersection point of the diagonals of $Q$ coincides with the midpoint of atleast one of the diagonals of $Q$. Our main result is that if $Q$ is a midpointdiagonal quadrilateral, then the smallest non-negative angle between equalconjugate diameters of $E_I$ equals the smallest non-negative angle between thediagonals of $Q$. This was proven in another earlier paper of the author forparallelograms.

Ellipses of minimal eccentricity inscribed in midpoint diagonal quadrilaterals