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Journal of Online Mathematics and its Applications
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The Most Marvelous Theorem in Mathematics
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.
Technologies Used in This Article
This article has several animated graphics that use Adobe Flash Player. Click on the link to download the plug-in.