3. Constructing a Point on a Cassini Oval
This Demonstration shows another ruler-and-compass construction of a point on a Cassini oval.[more]
An ellipse is given with the equation and eccentricity , . Choose any point on . Let be the point opposite and let be a point on different from and . Tangents to at and are parallel and meet the tangent at and at points and , respectively. Then .
Draw a circle with center and radius and a circle with center and radius ; suppose these meet in points and . But then . Thus is a point on a Cassini oval with foci and . The same is true for the point . It can be shown that the foci and are also on the oval.[less]
In Conics, Book III, theorem 42, Apollonius showed that for and .
 T. Heath, A History of Greek Mathematics, Volume II: From Aristarchus to Diophantus, New York: Dover Publications, 1981 p. 155.
 A. Ostermann and G. Wanner, Geometry by Its History, New York: Springer, 2012 pp. 76–78.