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An Elementary Proof of Marden's Theorem

by Dan Kalman

Award: Lester R. Ford

Year of Award: 2009

Publication Information: The American Mathematical Monthly, vol. 115, no. 4, April 2008, pp. 330–338.

Summary: Marden's Theorem deserves to be better known. That realization motivated Dan Kalman's lovely exposition of it. The theorem describes a thoroughly unexpected geometric connection between the roots of a cubic polynomial p with complex coefficients and the roots of the polynomial's derivative p'. To wit, if the roots of p are noncollinear points A, B, and C in the complex plane, the roots of p' are the foci of the unique ellipse inscribed in triangle ABC and tangent to the sides at their midpoints. Dan Kalman has provided an elementary, self-contained proof of Marden's Theorem.

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About the Author: Dan Kalman has been a member of the mathematics faculty at American University, Washington, DC since 1993. Prior to that he worked for eight years in the aerospace industry and taught at the University of Wisconsin, Green Bay. During the 1996-1997 academic year he served as an associate executive director of the MAA. Kalman has a B.S. from Harvey Mudd College and a Ph.D. from University of Wisconsin, Madison.

Subject classification(s): Numbers and Computation | Number Concepts | Complex Numbers | Geometry and Topology | Analytic Geometry | Calculus
Publication Date: 
Wednesday, September 2, 2009