Trisectrix

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Trisectrix

A trisectrix is a curve which is a variety of the Limaçon of Pascal, and named from its property of angle trisection. The polar equation is $r = 1 + 2cosθ$ .

To demonstrate the trisection property, draw a ray at an angle of $θ$ to the x-axis, beginning at point C (1,0) (the center of the small loop) which intersects the large loop of the trisectrix at point P. Draw another ray from the origin O (0,0) to point P. These two rays intersect at an angle of $\frac{1}{3}\theta$ .

To prove the trisection property, label point D (3,0) (the center of the large loop of the trisectrix), and draw a unit circle centered at C (1,0), whose polar equation is $r = 2cosθ$ . Label point Q where OP intersects the circle. Observe that $\triangle OCQ$ and $\triangle CQP$ are isosceles. (OQ and OP differ in length by 1, as seen from the two equations.) By the properties of isosceles triangles, $\angle OQC$ is twice $\angle OPC$ , and $\angle QCD$ is twice $\angle OQC$ . Thus $\angle PCD$ is three times $\angle OPC$ .