Trisectrix
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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 
.
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 
and 
are isosceles. (OQ and OP differ in length by 1, as seen from the two equations.) By the properties of isosceles triangles, 
is twice 
, and 
is twice . Thus 
is three times .
[edit] References
- This article incorporates text from the Encyclopædia Britannica Eleventh Edition, a publication now in the public domain.
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