A Generalization of the Bôcher-Grace Theorem

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The Bôcher-Grace Theorem can be stated as follows: Let p be a third degree complex polynomial. Then there is a unique inscribed ellipse interpolating the midpoints of the triangle formed from the roots of p, and the foci of the ellipse are the critical points of p. Here, we prove the following generalization: Let p be an nth degree complex polynomial and let its critical points take the form α+β kπ/n, k=1,...,n-1, β≠0. Then there is an inscribed ellipse interpolating the midpoints of the convex polygon formed by the roots of p, and the foci of this ellipse are the two most extreme critical points of p: α±β π/n.