4. Constructing a Point on a Cassini Oval
This Demonstration shows another construction of Cassini's oval. Start with the hyperbola with equation of eccentricity , . Select any point on . Let be the opposite point of and a point on different from and . The tangents on at and are parallel and meet the tangent at at points and , respectively. Then .[more]
Draw a circle with center and radius and a circle with center and radius ; suppose these meet in points and . But then . So 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 has shown that for and for the ellipse.
 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.