The Journal of Online Mathematics and Its Applications, Volume 8 (2008)

The Most Marvelous Theorem in Mathematics, Dan Kalman

Let `p`(`z`) be a third degree polynomial with complex coefficients, and whose roots `z`_{1}, `z`_{2}, and `z`_{3} are non-collinear points in the complex plane. Let `T` be the triangle with vertices at `z`_{1}, `z`_{2}, and `z`_{3}. There is a unique ellipse inscribed in `T` and tangent to the sides at their midpoints. The foci of this ellipse are the roots of `p`′(`z`).

The theorem is illustrated in Figure 1 below. The roots of `p` are at the vertices of the triangle, the midpoints of the sides of the triangle are indicated in red, and the foci of the ellipse are shown in blue. The foci are the roots of `p`′.

Figure 1: A Marvelous Theorem: `p`(`z`) = 0 at the vertices of the triangle; `p`′(`z`) = 0 at the foci of the inscribed ellipse.