Clairaut's Angle Trisection
This Demonstration shows Clairaut's angle trisection, which uses a hyperbola.
Let be the given angle with and on the circle with center . Let the points and trisect the segment and let the bisector of the angle cut at and the circle at . Then . Let the hyperbola with focus , vertex , and directrix cut the circle at point . Let be the orthogonal projection of on and let be the point symmetrical to relative to . Then .
 W. W. Rouse Ball, H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover Publications, 1987 p. 346.
"Clairaut's Angle Trisection"
Wolfram Demonstrations Project
Published: September 19 2012