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Marden's theorem - mklabgr - 06-17-2026
Marden's theorem
by Wikipedia
Summary
Marden’s theorem is a result in complex geometry and polynomial theory that describes a surprising connection between the roots of a cubic polynomial and the roots of its derivative. If the three roots of a cubic polynomial form the vertices of a triangle in the complex plane, then the two roots of the derivative are located at the two foci of the unique ellipse inscribed inside that triangle, known as the Steiner inellipse, which touches each side at its midpoint.
The theorem gives a precise geometric interpretation of the Gauss–Lucas theorem, showing not only that derivative roots lie inside the triangle formed by the original roots, but exactly where they are. Although named after mathematician Morris Marden, the result was first discovered earlier by Jörg Siebeck and has become a classic example of the deep relationship between algebra, geometry, and complex analysis.
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