Marden's Theorem

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Marden's theorem has been called the most marvelous theorem in mathematics (see the reference in the Details section below). It reads: Given any complex cubic polynomial whose roots form a triangle in the complex plane, there is a unique ellipse inscribed in this triangle that is tangent to the sides of the triangle at their midpoints. The roots of the derivative are the foci of this ellipse.


In this Demonstration the roots of (the vertices of the triangle), are locators. Drag them and watch the ellipse and its foci move.


Contributed by: Bruce Torrence (April 2008)
Open content licensed under CC BY-NC-SA



The Demonstration works as follows: The polynomial is differentiated and the roots of its derivatives are plotted as orange points. The ellipse with these points as its foci and passing through the midpoint of one side of the triangle is then plotted. The fact that this ellipse is inscribed in the triangle, with points of tangency being the midpoints of all three sides, illustrates the theorem.

For details see Dan Kalman's article, "The Most Marvelous Theorem in Mathematics", in Journal of Online Mathematics and Its Applications. Further references can be found there.

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