# Marden's Theorem - a marvelous theorem in mathematics

- Author:
- Irina Boyadzhiev

- Topic:
- Mathematics

Let be a third-degree polynomial with complex coefficients, whose roots , , and are non-collinear points in the complex plane. Let be the triangle with vertices , , and . There is a unique ellipse inscribed in and tangent to the sides at their midpoints. The theorem says that the foci of this ellipse are the roots of .
The applet below demonstrates Marden’s theorem for polynomials with

**real**coefficients.- Click on the “Roots of f(x) checkbox to show the roots of the polynomial.
- Show the first derivative and the roots of the first derivative.
- Sow the ellipse with foci at the roots of the first derivative and passing through the midpoints of the sides of the triangle

- Show the second derivative f ‘’ (x) and its roots for the demonstration of the Gauss–Lucas theorem in the real case.
- Drag the sliders to change the coefficients of the polynomial.