Ellipses Inscribed in Parallelograms

TL;DR

This paper investigates the properties of ellipses inscribed in parallelograms and rectangles, establishing uniqueness, geometric relationships, and conditions for bielliptic parallelograms with minimal eccentricity.

Contribution

It proves the uniqueness of minimal eccentricity inscribed ellipses in parallelograms and rectangles, and characterizes bielliptic parallelograms through diagonal and side length relations.

Findings

Unique minimal eccentricity inscribed ellipse in parallelograms and rectangles.

The smallest angle between conjugate diameters equals the smallest angle between diagonals.

Bielliptic parallelograms satisfy a specific relation between diagonals and sides.

Abstract

We prove that there exists a unique ellipse of minimal eccentricity, E_{I}, inscribed in a parallelogram, D. We also prove that the smallest nonnegative angle between equal conjugate diameters of E_{I} equals the smallest nonnegative angle between the diagonals of D. We also prove that if E_{M} is the unique ellipse inscribed in a rectangle, R, which is tangent at the midpoints of the sides of R, then E_{M} is the unique ellipse of minimal eccentricity, maximal area, and maximal arc length inscribed in R. Let D be any convex quadrilateral. In previous papers, the author proved that there is a unique ellipse of minimal eccentricity, E_{I}, inscribed in D, and a unique ellipse, E_{O}, of minimal eccentricity circumscribed about D. We defined D to be bielliptic if E_{I} and E_{O} have the same eccentricity. In this paper we show that a parallelogram, D, is bielliptic if and only if the square of the length of one of the diagonals of D equals twice the square of the length of one of the sides of D.