The chordal condition and Stewart’s theorem

In the present paper we consider two families of curves, namely the family of all ellipses and the family of all Cassini ovals. We introduce the notion of a chordal condition of curve at a fixed point. The chordal condition is a generalization of a well-known property of ellipses, from their foci to other points. This paper is based on Stewart’s theorem. A new definition of a Cassini oval is given. A forthcoming paper will contain further extensions of results given here, including other families of curves.