Editing Parabola (section)

== As trisectrix ==
[[File:Angle trisection with parabola2.svg|thumb|upright=1.25|Angle trisection with a parabola]]
A parabola can be used as a [[trisectrix]], that is it allows the [[Angle trisection|exact trisection of an arbitrary angle]] with straightedge and compass. This is not in contradiction to the impossibility of an angle trisection with [[compass-and-straightedge construction]]s alone, as the use of parabolas is not allowed in the classic rules for compass-and-straightedge constructions.

To trisect <math>\angle AOB</math>, place its leg <math>OB</math> on the ''x'' axis such that the vertex <math>O</math> is in the coordinate system's origin. The coordinate system also contains the parabola <math>y = 2x^2</math>. The unit circle with radius&nbsp;1 around the origin intersects the angle's other leg <math>OA</math>, and from this point of intersection draw the perpendicular onto the ''y'' axis. The parallel to ''y'' axis through the midpoint of that perpendicular and the tangent on the unit circle in <math>(0, 1)</math> intersect in <math>C</math>. The circle around <math>C</math> with radius <math>OC</math> intersects the parabola at <math>P_1</math>. The perpendicular from <math>P_1</math> onto the ''x'' axis intersects the unit circle at <math>P_2</math>, and <math>\angle P_2OB</math> is exactly one third of <math>\angle AOB</math>.

The correctness of this construction can be seen by showing that the ''x'' coordinate of <math>P_1</math> is <math>\cos(\alpha)</math>. Solving the equation system given by the circle around <math>C</math> and the parabola leads to the cubic equation <math>4x^3 - 3x - \cos(3\alpha) = 0</math>. The [[List of trigonometric identities#Triple-angle formulae|triple-angle formula]] <math>\cos(3\alpha) = 4 \cos(\alpha)^3 - 3 \cos(\alpha)</math> then shows that <math>\cos(\alpha)</math> is indeed a solution of that cubic equation.

This trisection goes back to [[René Descartes]], who described it in his book {{lang|fr|La Géométrie}} (1637).<ref>{{cite journal |first=Robert C. |last=Yates |title=The Trisection Problem |journal=National Mathematics Magazine |volume=15 |issue=4 |year=1941 |pages=191–202 |doi=10.2307/3028133 |jstor=3028133 }}</ref>