Midpoint Diagonal Quadrilaterals

TL;DR

This paper characterizes midpoint diagonal quadrilaterals through properties of inscribed ellipses, proving that such quadrilaterals have ellipses with specific tangent and conjugate diameter properties, including a unique minimal eccentricity inscribed ellipse.

Contribution

It establishes new geometric characterizations of midpoint diagonal quadrilaterals based on inscribed ellipses, extending properties known for parallelograms.

Findings

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

Each inscribed ellipse has a unique pair of conjugate diameters parallel to the diagonals.

There exists a unique inscribed ellipse with minimal eccentricity in such quadrilaterals.

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