Steiner Formula for Cassini Oval

A Cassini oval is the locus of points such that , where and . If the foci and , then
,
.
For the normal vector at a point on the oval,
,
where is the unit vector in the direction of .
So the normal is the diagonal of the parallelogram obtained by prolonging the vector by the magnitude of and by the magnitude of . The tangent is the line through that is perpendicular to the normal.
Let be the intersection of the perpendicular to at and the tangent. Let be the intersection of the perpendicular to at and the tangent.
Apply the law of sines to the triangle to get Steiner's formula (1835)
.
This can be applied to show that is the midpoint of the segment : the angles at and at are and , respectively, as angles with orthogonal legs. So .

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Reference
[1] A. Ostermann and G. Wanner, Geometry by Its History, New York: Springer, 2012.
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