Ellipses of minimal eccentricity inscribed in midpoint diagonal quadrilaterals

Abstract

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.

There is an error in Remark 4 following Theorem 3. It turns out that in the example I gave for Q, Q is a type 2 midpoint diagonal quadrilateral. Indeed, I can now prove that if Q is both tangential and orthodiagonal, then Q is a midpoint diagonal quadrilateral