When is an ellipse inscribed in a quadrilateral tangent at the midpoint of two or more sides ?

Abstract

In "Quartic Coincidences and the Singular Value Decomposition" by Clifford and Lachance, Mathematics Magazine, December, 2013, it was shown that if there is a midpoint ellipse(an ellipse inscribed in a quadrilateral, $Q$, which is tangent at the midpoints of all four sides of $Q$), then $Q$ must be a parallelogram. We strengthen this result by showing that if $Q$ is not a parallelogram, then there is no ellipse inscribed in $Q$ which is tangent at the midpoint of three sides of $Q$. Second, the only quadrilaterals which have inscribed ellipses tangent at the midpoint of even two sides of $Q$ are trapezoids or what we call a midpoint diagonal quadrilateral(the intersection point of the diagonals of $Q$ coincides with the midpoint of at least one of the diagonals of $Q$).