An Elementary Proof of Marden's Theorem

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 Bôcher [1]. In his presentation Marden states the theorem in a more general form than given above, corresponding to the logarithmic derivative of a product (z − z1)1(z − z2)2(z − z3)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 book, is incomplete for reasons that will be made clear below. A closely related argument in Bôcher’s paper is also incomplete, although in a different way. By combining the two arguments, a complete proof of Marden’s Theorem is obtained. Moreover, the proof is completely elementary, requiring very little beyond standard topics from undergraduate mathematics. To be honest, there are rather a lot of these topics required, spanning analytic geometry, linear algebra, complex analysis, calculus, and properties of polynomials. To me, the way all of these topics weave together is part of the charm of the theorem, and presenting the proof is the primary motivation for this paper. In an on-line paper [3] a more completely self-contained exposition is provided, with hypertext links to discussions of many of the necessary background topics, as well as animated graphics dramatizing some of the geometric ideas of the proof. Before proceeding further, some additional observations may be illuminating. First, it is possible that the unique inscribed ellipse mentioned in the theorem is actually a circle. In this case the foci coincide, indicating that p′(z) has a double root. This case can only occur if the circumscribing triangle is equilateral. In fact, this special case is easy to verify by assuming that p′(z) has a double root, and deducing the form of p. The roots of p are then seen to be vertices of an equilateral triangle centered at the repeated root of p′.