4739 WordsFeb 2, 201419 Pages

An Elementary Proof of Marden’s Theorem
Dan Kalman
What I call Marden’s Theorem is one of my favorite results in mathematics. It establishes a fantastic relation between the geometry of plane ﬁgures and the relative positions of roots of a polynomial and its derivative. Although it has a much more general statement, here is the version that I like best: Marden’s Theorem. Let p(z) be a third-degree polynomial with complex coefﬁcients, and whose roots z 1 , z 2 , and z 3 are noncollinear points in the complex plane. Let T be the triangle with vertices at z 1 , z 2 , and z 3 . There is a unique ellipse inscribed in T and tangent to the sides at their midpoints. The foci of this ellipse are the roots of p (z). I call this Marden’s Theorem because I ﬁrst read it in M. Marden’s wonderful book [6]. But this material appeared previously in Marden’s earlier paper [5]. In both sources Marden attributes the theorem to Siebeck, citing a paper from 1864 [8]. Indeed, Marden reports appearances of various versions of the theorem in nine papers spanning the period from 1864 to 1928. Of particular interest in what follows below is an 1892 paper by Maxime Bˆ cher [1]. o In his presentation Marden states the theorem in a more general form than given above, corresponding to the logarithmic derivative of a product (z − z 1 )m 1 (z − z 2 )m 2 (z − z 3 )m 3 where the only restriction on the exponents m j is that they be nonzero, and with a general conic section taking the place of the ellipse. For this discovery he credits Linﬁeld [4], who obtained it as a corollary to an even more general result “established by the use of line coordinates and polar forms.” Marden asserts the desirability of a more elementary proof, and proceeds to give one based on the optical properties of conic sections. Interestingly, Marden’s proof, which appears in basically the same form in both his paper and his

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