An Area Inequality for Ellipses Inscribed in Quadrilaterals

TL;DR

This paper establishes an area inequality for ellipses inscribed in convex quadrilaterals, extending known triangle results, and explores properties of the maximal inscribed ellipse in parallelograms.

Contribution

It proves an area ratio bound for inscribed ellipses in convex quadrilaterals and characterizes when equality occurs, extending classical triangle results.

Findings

Area ratio of inscribed ellipse to quadrilateral ≤ pi/4

Equality holds iff the quadrilateral is a parallelogram and ellipse tangent at midpoints

Foci of maximal inscribed ellipse in parallelogram lie on the orthogonal least squares line

Abstract

If E is any ellipse inscribed in a convex quadrilateral, D, then we prove that Area(E)/Area(D) is less than or equal to pi/4, and equality holds if and only if D is a parallelogram and E is tangent to the sides of D at the midpoints. This extends well known results for ellipses inscribed in triangles. We also prove that the foci of the unique ellipse of maximal area inscribed in a parallelogram, D, lie on the orthogonal least squares line for the vertices of D. This does not hold in general for convex quadrilaterals.