Marden's Theorem

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

p(z)

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

p'(z)

are the foci of this ellipse.

In this Demonstration the roots of

p(z)

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