Marden's theorem
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In mathematics, Marden's theorem, named after Morris Marden, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative:
- Suppose the zeroes z1, z2, z3 of a third-degree polynomial
are non-colinear. There is a unique ellipse inscribed in the triangle with vertices z1, z2, z3 and tangent to the sides at their midpoints: the Steiner inellipse. The foci of that ellipse are the zeroes of the derivative 
.
Marden attributes the theorem to Jörg Siebeck[1] and cites nine papers that included version of the theorem.
[edit] References
- ^ Siebeck, Jörg (1864), “Über eine neue analytische Behandlungweise der Brennpunkte”, Journal für die reine und angewandte Mathematik 64: 175–182, ISSN 0075-4102
- Kalman, Dan (April 2008), “An Elementary Proof of Marden's Theorem”, The American Mathematical Monthly 115: 330–338, ISSN 0002-9890
- Kalman, Dan (April 2008), “The Most Marvelous Theorem in Mathematics”, Journal of Online Mathematics and its Applications
- Marden, Morris (1945), “A note on the zeroes of the sections of a partial fraction”, Bulletin of the American Mathematical Society 51 (12): 935–940, ISSN 0002-9904, <http://www.ams.org/bull/1945-51-12/S0002-9904-1945-08470-5/home.html>
- Marden, Morris (1966), Geometry of Polynomials, Mathematical Surveys, number 3, Providence, R.I.: American Mathematical Society
