3. Constructing a Point on a Cassini Oval

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This Demonstration shows another ruler-and-compass construction of a point on a Cassini oval.


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.


Contributed by: Marko Razpet and Izidor Hafner (July 2018)
Open content licensed under CC BY-NC-SA



In Conics, Book III, theorem 42, Apollonius showed that for and .


[1] T. Heath, A History of Greek Mathematics, Volume II: From Aristarchus to Diophantus, New York: Dover Publications, 1981 p. 155.

[2] A. Ostermann and G. Wanner, Geometry by Its History, New York: Springer, 2012 pp. 76–78.

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