A generalization of parallelograms involving inscribed ellipses, conjugate diameters, and tangency chords

TL;DR

This paper characterizes inscribed ellipses in midpoint diagonal quadrilaterals, extending properties known for parallelograms, and identifies a unique minimal eccentricity ellipse with conjugate diameters aligned with the diagonals.

Contribution

It generalizes properties of inscribed ellipses from parallelograms to midpoint diagonal quadrilaterals and proves the existence of a unique minimal eccentricity ellipse with specific conjugate diameter orientations.

Findings

Ellipses inscribed in midpoint diagonal quadrilaterals have tangency chords parallel to diagonals.

Inscribed ellipses have conjugate diameters parallel to diagonals.

There exists a unique minimal eccentricity inscribed ellipse with conjugate diameters parallel to diagonals.

Abstract

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 pair of conjugate diameters parallel to the diagonals of $Q$. Finally, we show that there is a unique ellipse, $E_I$, of minimal eccentricity incribed in a midpoint diagonal quadrilateral, $Q$, and we show that the equal conjugate diameters of $E_I$ are parallel to the diagonals of $Q$.