## 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.

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