The Journal of Online Mathematics and Its Applications, Volume 8 (2008)

The Most Marvelous Theorem in Mathematics, Dan Kalman

I use the name Marden's Theorem because I first read it in Morris Marden's wonderful book [7]. While Marden attached no particular name to the result, he was a careful scholar and traced its history through several variations back to an 1864 paper of J. Siebeck.

On this point I have sadly failed to live up to Marden's scholarly standard. The intriguing result about the roots of a cubic remained in my memory long after I forgot the names of Siebeck and others Marden cited. Over the years, I often mentioned the theorem to colleagues. Partly as a matter of convenience, and partly because so marvelous a result deserves to have a special name, I started to refer to it as Marden's Theorem. I now know that this is a misattribution, but it is an old habit that is hard to break. It is well known that many famous theorems are named not for their original discoverers, but for later investigators in whose work the theorems were applied, extended, or popularized. Perhaps my perpetuation of a historically inaccurate label for the theorem in this presentation can be forgiven as just one more example of this phenomenon.

These introductory remarks touch on two themes: the history of the result as documented in Marden's scholarship; and my own less respectable history of interest in the result. These themes will be discussed at greater length in the two sections below.

Marden's presentation in his book [7] essentially repeats material that appeared previously in an earlier paper [6]. The attribution to Siebeck occurs in both places, citing the 1864 paper [9]. Indeed, Marden reports appearances of various versions of the theorem in nine papers spanning the period from 1864 to 1928. One reference that has particular interest is an 1892 paper by Maxime Bôcher [1]. This paper inspired part of the proof given in this web presentation. Also, Bôcher's name is familiar to modern mathematicians in connection with a prize given in his honor by the American Mathematical Society

In Marden's presentation he states the theorem in a more general form than in my statement. Marden's version corresponds to the logarithmic derivative of a product `f`(`z`) = (`z` − `z`_{1})^{m1}(`z` − `z`_{2})^{m2}(`z` − `z`_{3})^{m3}. The natural logarithm of the product is given by

log `f`(`z`) = `m`_{1} log(`z` − `z`_{1}) + `m`_{2} log(`z`
− `z`_{2}) + `m`_{3} log(`z` − `z`_{3}),

and has derivative

On the other hand, we know from the chain rule that (log `f`(`z`)) ′ = `f` ′(`z`) `/ f`(`z`). This shows that the roots of `f` ′(`z`) which are not also roots of `f`(`z`) must be roots of (log` f`(`z`))′. So, finding the roots of `f`
′(`z`) is equivalent to finding the roots of (log` f`(`z`))′, provided `f`(`z`) has no repeated roots.

Notice that `f`(`z`) is a polynomial when the exponents `m _{j}` are nonnegative integers. But this restriction is not assumed in the context of Marden's version of the theorem. He assumes only that the

Why did Marden particularize Linfield's result? He wanted to give a more accessible proof. Marden points out that the most general version was "*established by the use of line coordinates and polar forms.*" He goes on to say "*In view, however, of the elementary character of Theorem 1, it seems desirable to furnish for it an elementary proof based upon some familiar property of the conics.*" The proof he then provides uses the optical properties of conics, and in the particular case of positive exponents `m`_{j}, the optical property of ellipses.

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. The proof that I give in this web presentation is a combination of the arguments of Marden and Bôcher.

One interesting aspect of reading papers which date to the late 19th and early 20th centuries is the background knowledge that the authors assume. For example, in Bôcher's paper, he refers to the *maximum ellipse* of a triangle, meaning the inscribed ellipse tangent at the midpoints of the sides, as if that terminology were common knowledge. He takes it for granted that this ellipse is uniquely determined. He also states as self evident the observation that "*only one conic can be drawn with a given point as centre and tangent to three given lines*." None of these were familiar to me, and I speculate they could no longer be considered common knowledge for general mathematicians. Note that the uniqueness of the inscribed ellipse is an important aspect of Marden's Theorem as I have stated it here. Without uniqueness, inscribing an ellipse cannot be considered to lead to a specific set of foci, leaving some ambiguity about which ellipse might lead to the roots of `p`′.

For the sake of clarity, I will describe the logical gap in Marden's argument in the context of the theorem as I have stated it, corresponding to `m`_{j} = 1 for `j` = 1,2,3, in his more general statement of the result. Accordingly, let the polynomial `p`(`z`), its roots `z`_{1}, `z`_{2}, `z`_{3}, and the triangle `T` be as in my statement of Marden's Theorem. Marden shows that an ellipse with foci at the roots of `p`′ and which passes through the midpoint of one side of `T` is actually tangent to the side there. By symmetry, this argument applies to any side of the triangle. That is as far as Marden's proof goes. The appeal to symmetry establishes the existence of three ellipses, each with foci at the roots of `p`′, and each tangent to one side of the triangle at its midpoint. To complete the proof, it must be shown that these three ellipses are actually one and the same.

Regarding Bôcher's proof, the only gap is not in the logic, but rather in my background knowledge. He shows that under the circumstances just described, an ellipse with foci at the roots of `p`′ and which is tangent to one side of the triangle must also be tangent to the other two sides of the triangle. This is where he makes his observation about the existence of only one ellipse with a given center and tangent to three given lines. Since we already know of the existence of the *maximal* ellipse, he is able to show that it must coincide with the one having roots of `p`′ as its foci. This provides a complete proof of Marden's theorem, provided one already knows the uniqueness result concerning tangency to three lines. For those like me who are not familiar with this result, an alternate proof can be obtained using Marden's approach.

I first encountered Marden's theorem in my third year in college, while reading from Marden's book [7]. It struck me as the most amazing combination of ideas, and intrigued me. But I could not make much sense of the proof. That is so long ago now, that I have no clear recollection about what part of the proof confused me. But I don't think I noticed any logical gap, and I certainly did not come away with the feeling that I understood *why* the result holds.

Over many years, I never forgot the statement of Marden's theorem, but I also never revisited the proof. Then, just a few years ago, while studying something else, I learned a related fact about polynomials: the average of the roots of a polynomial is the same as the average of the roots of the derivative. This is easy to prove and to understand (click here for details). Let's apply this fact to the cubic polynomial of Marden's theorem. The three roots are the vertices of the triangle `T`, so the average of these roots is the centroid of `T`. We now know that the centroid must also be the average of the roots of `p`′. So, if you specify a trial location for one root of `p`′, reflection through the centroid determines the required location for the other root.

At the time all this occurred to me, I was at a workshop connected with the MAA's Project WELCOME, working with computer software called Mathwright. The point of Mathwright is to create interactive dynamic computer environments dramatizing mathematical relationships and ideas. It was only natural then for me to think of a dynamic portrayal of Marden's theorem. The animated dramatization included in this web presentation is a demonstration from the Mathwright portrayal I implemented at the workshop. However, I knew that to give a full account of the theorem, I needed to understand the proof. Consequently, after a hiatus of nearly thirty years, I returned to Marden's book, and finally understood all the steps of his proof. That was when I first noticed the gap in his proof, and started to read the other references.