Magic conics, their integer points and complementary ellipses

Abstract

The aim of this paper is to introduce and study the class of conics provided by the symmetric matrices of the 3 × 3 magic squares. This class depends on three real parameters and various relationships between these parameters give special subclasses of conics. Although there are no magic circles we find an ellipse, a parabola and two hyperbolas of magic type. A search of integer points and a complex approach are also included. We study also a pair of complementary ellipses, called Pythagorean.