Le $p\in\Bbb R[X]$ be a 3rd degree polynomial. Suppose it has one real root and two complex conjugate roots: these three points forms a triangle in the complex plane.
Consider the ellipse inscribed in the triangle, touching the three mid-point of the three sides of the triangle (why the conic individuated by these three point is an ellipse? is this always true?).
Well, the two focal points of this ellipise, are the roots of $p'$.
How can I prove it?