A Geometric Proof of the Siebeck–Marden Theorem

The Siebeck–Marden theorem relates the roots of a third degree polynomial and the roots of its derivative in a geometrical way. A few geometric arguments imply that every inellipse for a triangle is uniquely related to a certain logarithmic potential via its focal points. This fact provides a new direct proof of a general form of the result of Siebeck and Marden. Given three noncollinear points a, b, c ∈ C, we can consider the cubic polynomial P(z) = (z − a)(z − b)(z − c), whose derivative P ′(z) has two roots f1, f2. The Gauss–Lucas theorem is a well-known result which states that given a polynomial Q with roots z1, . . . , zn , the roots of its derivative Q ′ are in the convex hull of z1, . . . , zn . In the simple case where we have only three roots, there is a more precise result. The roots f1, f2 of the derivative polynomial are situated in the interior of the triangle abc and they have an interesting geometric property: f1 and f2 are the focal points of the unique ellipse that is tangent to the sides of the triangle abc at its midpoints. This ellipse is called the Steiner inellipse associated to the triangle abc. In the rest of this note, we use the term inellipse to denote an ellipse situated in a triangle that is tangent to all three of its sides. This geometric connection between the roots of P and the roots of P ′ was first observed by Siebeck (1864) [12] and was reproved by Marden (1945) [8]. There has been substantial interest in this result in the past decade: see [3],[5, pp. 137–140] [7],[9],[10],[11]. Kalman [7] called this result Marden’s theorem, but in order to give credit to Siebeck, who gave the initial proof, we call this result the Siebeck–Marden theorem in the rest of this note. Apart from its purely mathematical interest, the Siebeck–Marden theorem has a few applications in engineering. In [2] this result is used to locate the stagnation points of a system of three vortices and in [6] this result is used to find the location of a noxious facility location in the three-city case. The proofs of the Siebeck–Marden theorem found in the references presented above are either algebraic or geometric in nature. The initial motivation for writing this note was to find a more direct proof, based on geometric arguments. The solution was found by answering the following natural question: Can we find two different inellipses with the same center? Indeed, let’s note that (a + b + c)/3 = ( f1 + f2)/2, which means that the centers of ellipses having focal points f1, f2 coincide with the centroid of the triangle abc. The geometric aspects of the problem can be summarized in the following questions. 1. Is an inellipse uniquely determined by its center? 2. Which points in the interior of the triangle abc can be centers of an inellipse? 3. What are the necessary and sufficient conditions required such that two points f1 and f2 are the focal points of an inellipse? 4. Is there an explicit connection between the center of the inellipse and its tangency points? http://dx.doi.org/10.4169/amer.math.monthly.124.5.459 MSC: Primary 97G50