2. Normal and Tangent to a Cassini Oval

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This Demonstration shows how to construct the normal and tangent to a Cassini oval at a point .

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


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 .


Snapshots



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