Additional Relations Between Root Locations and The Steiner Inellipse
By the Gauss–Lucas theorem, the root of the double derivative p"(z) must be the average of the two foci, which is the center point of the ellipse and the centroid of the triangle. In the special case that the triangle is equilateral (as happens, for instance, for the polynomial p(z) = z3 − 1) the inscribed ellipse degenerates to a circle, and the derivative of p has a double root at the center of the circle. Conversely, if the derivative has a double root, then the triangle must be equilateral (Kalman 2008a).
Read more about this topic: Marden's Theorem
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