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Descartes used the intersection of a circle and a parabola to trisect an angle. The equations of the circle and parabola are

and

, respectively. The coordinates of the intersection satisfy

. Since

, by taking

, the smaller positive root of the last equation is

Reference

[1] W. W. Rouse Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover Publications, 1987 p. 346.

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Descartes's Angle Trisection