The Most Marvelous Theorem in Mathematics

Dan Kalman

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Marden's Theorem concerns the relative positions of the roots of a cubic polynomial and those of its derivative. Specifically, if the cubic has distinct non-collinear roots in the complex plane, and thus are the vertices of a triangle T, then the roots of the derivative are the foci of the unique ellipse inscribed in T and tangent to the sides at their midpoints. The theorem has an elementary proof, but it draws on background knowledge from a wide array of topics in the undergraduate curriculum. This paper provides a completely self-contained proof of Marden's theorem, along with dynamic geometry animations and some of the history of the result.

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Publication Data

  • Published April 2008
  • Copyright © 2008 by Dan Kalman

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