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Trisecting an Angle with a Limaçon

The graph of a limaçon in polar coordinates is ; it can be constructed geometrically. Let be a ray with polar angle . Produce this line and determine point the so that . Then .
A limaçon with can be used to trisect angles. Choose the angle so that one leg intersects the limaçon at .
Since the triangle is isosceles, . Since the triangle is also isosceles, . So , or .

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Isaac Newton used a limaçon to trisect an angle.
Reference
[1] T. Iwamoto. "Trisection Using Special Curves." (Nov 22, 2006) www.takayaiwamoto.com/Greek_Math/Trisect/Special_Curves/Special_Curves_Tri.html.
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