Ellipses of minimal eccentricity inscribed in midpoint diagonal quadrilaterals

TL;DR

This paper proves the uniqueness of the minimal eccentricity inscribed ellipse in midpoint diagonal quadrilaterals and relates its conjugate diameters' angles to the diagonals' angles, extending previous results from parallelograms.

Contribution

It establishes the existence and uniqueness of the minimal eccentricity inscribed ellipse in midpoint diagonal quadrilaterals and links its properties to the quadrilateral's diagonals.

Findings

Unique minimal eccentricity inscribed ellipse exists in midpoint diagonal quadrilaterals.

The smallest angle between equal conjugate diameters of this ellipse equals the smallest diagonal angle.

Extension of previous parallelogram results to a broader class of 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.