Marden's Theorem - a marvelous theorem in 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
In addition, the Gauss–Lucas theorem states that the root of the second  derivative must be the midpoint of the two foci.
  • 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.
See "An Elementary Proof of Marden's Theorem" by Dan Kalman, American Mathematical Monthly, Volume 115, Number 4, April 2008, pp 330-338.