2. Normal and Tangent to a Cassini Oval

This Demonstration shows how to construct the normal and tangent to a Cassini oval at a point .
A Cassini oval is the locus of points such that , where and . If the foci and , then
,
.
Extend the segment to so that is the midpoint of . Let . Construct a point so that . The line is normal to the oval at and the tangent is the perpendicular to at .

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DETAILS

Let , let be the angle between and the normal to the oval at , and let be the angle between the normal and . We must prove that and .
Since is an external angle of the triangle , . Suppose .
Using the Steiner formula
,
(.
On the other hand, by the tangent law for the triangle ,
.
Since
, .
Thus and .
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