Let [math]p(z)[/math] be a third-degree polynomial with complex coefficients, whose roots [math] z_1[/math], [math]z_2[/math], and [math]z_3[/math] are non-collinear points in the complex plane. Let [math]T[/math] be the triangle with vertices [math]z_1[/math], [math]z_2[/math], and [math]z_3[/math]. There is a unique ellipse inscribed in [math]T[/math] and tangent to the sides at their midpoints. The theorem says that the foci of this ellipse are the roots of[math] p′(z)[/math]. The applet below demonstrates Marden’s theorem for polynomials with [b]real[/b] coefficients. [list] [*]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 [/list] In addition, the Gauss–Lucas theorem states that the root of the second derivative must be the midpoint of the two foci. [list][*]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. [/list]
See "An Elementary Proof of Marden's Theorem" by Dan Kalman, American Mathematical Monthly, Volume 115, Number 4, April 2008, pp 330-338.