Focal Circular Cubic and Complex Numbers

A focal circular cubic is a locus of the foci of conics tangent to a given quadrilateral. In this article, we derive some of the focal curve’s basic properties. We study the curve in the complex plane and prove that the complex coordinates of pairs of foci satisfy a quadratic equation. This equation can be expressed as a linear combination of two basic quadratic equations, which form a basis of a vector space. Furthermore, we give a nonstandard analytical condition, expressed in complex numbers, under which a circle can be inscribed in a quadrilateral. Finally, we leave the complex plane and show the construction of an arbitrary pair of foci of the curve by Euclidean means.