An Area Inequality for Ellipses Inscribed in Quadrilaterals

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.

Submitted for publication to the Journal of Mathematical Inequalities. 16 pages, no figures