Marden's theorem

Remarks

The ellipse is the Steiner inellipse of $T$. If $p$ is monic, then $p'(z) = (z - z_1)(z - z_2) + (z - z_2)(z - z_3) + (z - z_3)(z - z_1)$. One proof computes the foci of the Steiner inellipse and checks, with the reflection property of the tangent lines of an ellipse, that they are the roots of this quadratic. For an equilateral triangle the inellipse is the incircle, and for $p(z) = z^3 - 1$ its centre $0$ is the double root of $p'(z) = 3z^2$. Example: for $p(z) = z(z - 1)(z - i)$, one gets $p'(z) = 3z^2 - 2(1 + i)z + i$, whose roots are the foci of the inellipse of the triangle $0, 1, i$. The theorem refines the Gauss–Lucas theorem for cubics: the critical points lie not only in the triangle but at the foci of a specific ellipse inside it.

Going deeper

For a student

The roots of $p'$ are the critical points of $p$, so the theorem locates the critical points of a cubic from its roots alone. It is a classical result of the geometry of polynomials, which studies where the zeros of a polynomial and of its derivatives lie. Together with Sendov's conjecture, it belongs to the study of how the zeros of $p'$ sit relative to the zeros of $p$.