Cubics, hyperbolas and Steiner ellipses (Q2051397)

This article looks at the surprising connections between the algebraically defined set of solutions of a polynomial \(f\) of degree \(3\), and purely geometric entities, such as hyperbolas, the Steiner inellipse and the circumscribed Steiner ellipse associated with the triangle formed in the complex plane by the non-collinear vertices of \(f\), in the spirit of the Siebeck-Marden theorem. The connection with hyperbolas is valid for all cubics, regardless of the geometric structure of the complex roots.