This Demonstration shows another ruler-and-compass construction of a point on a Cassini oval.
Let and be two fixed points (the foci) a distance apart. A Cassini oval (or Cassini ellipse) is a quartic curve traced by a point such that the product of the distances is a constant .
Let and let be the circle with center and radius . Let be a point on and let be the midpoint of . Let be the orthogonal projection of on the perpendicular bisector of . Let be the circle with center and radius ; let be the circle with center and radius . Let be the intersection of the circles and . Let be the angle between and .
Then , . The difference of these two equations gives , so that satisfies the defining condition for the oval.