Trisectrix: An Angle Trisection Curve

The Trisectrix is the curve, in green, with polar equation r = 1 + 2 cos(θ)

Angle PCD is trisected by the curve, because the measure of angle OPC is one third that of PCD.

Proof

We will denote the measure of angle OPC by α, and show that PCD has a measure of 3α.  Begin by drawing OP, intersecting the red circle at Q.  Note that OCQ is isosceles, because CO and CQ are radii of the red circle.  Note, too, that CQP is isosceles, because the rays OQ and OP differ in length by 1 (note their equations to see why), and so QP=1.

Now in triangle CQP, we see if QCP=α, then QPC=α as well, so the external angle OQC=2α.

In triangle OCQ, we see since OQC=2α, QOC=2α, and so its external angle QCD is 4α, so the measure of PCD is 3α.

Internet References

Xah Lee's Visual Dictionary of Famous Plane Curves: Trisectrix 

Wikipedia: Trisectrix 

Related pages in this website:

Back to the Geometry and Trig home, or Triangles and Polygons 

Law of Sines - Given triangle ABC with opposite sides a, b, and c, a/(sin A) = b/(sin B) = c/(sin C) = the diameter of the circumscribed circle.

Circumscribed Circle - The radius of a circle circumscribed around a triangle is R = abc/(4K), where K is the area of the triangle.

Inscribed Angle -- proof that an angle inscribed in a circle is half the central angle that is subtended by the same arc

Triangle Trisection -- If a point, P, on the median of triangle ABC is the isogonal conjugate of point Q, on the altitude of ABC, then ABC is a right triangle.

The webmaster and author of the Math Help site is Graeme McRae.
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