The locus of centers of ellipses inscribed in quadrilaterals

Abstract

Let R be a four-sided convex polygon in the xy plane and let M1 and M2 be the midpoints of the diagonals of R. It is well-known that if E is an ellipse inscribed in R, then the center of E must lie on Z, the open line segment connecting M1 and M2. We prove the converse: If (h,k) lies in Z, then there is a unique ellipse with center (h,k) inscribed in R. This completely characterizes the locus of centers of ellipses inscribed in R. This result was proved by the author in ("Finding ellipses and hyperbolas tangent to two, three, or four given lines", Southwest Journal of Pure and Applied Mathematics 1(2002), 6-32), but the approach given here is decidedly different and much shorter and more succinct. In addition, we are also able to prove that there is a unique ellipse of maximal area inscribed in R. The approach in this paper is to use a theorem of Marden relating the foci of an ellipse tangent to the lines thru the sides of a triangle and the zeros of a partial fraction expansion.