View source for Trisection of an angle

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The problem of dividing an angle into three equal parts.

The special case of trisection using only ruler-and-compass construction was one of the classical problems of Antiquity. The solution of the problem of trisecting an angle $\phi$ reduces to finding rational roots of a cubic equation $4x^3-3x-\cos\phi=0$, where $x=\cos(\phi/3)$, which, in general, is not solvable by quadratic radicals: that is, the roots of the general cubic do not lie in the field of [[constructible number]]s. Thus, the problem of trisecting a general angle cannot be solved by means of ruler and compass, as was proved in 1837 by P. Wantzel. However, such a construction is possible for angles $m\cdot90^\circ/2^n$, where $n,m$ are integers.

The problem may be solved by using other means and instruments of construction (for example, the [[Dinostratus quadratrix]] or the [[conchoid]]).

====References====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.I. Manin,   "Ueber die Lösbarkeit von Konstruktionsaufgaben mit Zirkel und Lineal" , ''Enzyklopaedie der Elementarmathematik'' , '''4. Geometrie''' , Deutsch. Verlag Wissenschaft.  (1969)  pp. 205–230  (Translated from Russian)</TD></TR>
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====Comments====
The problem of trisection of an angle, like [[duplication of the cube]], is one of the problems dealt with in [[Galois theory]], cf. also [[#References|[a3]]].

A remarkable result on trisection of the angles of a triangle is F. Morley's theorem (1899), stating that the three points of intersection of the adjacent trisectors of the angles of an arbitrary triangle form an equilateral triangle (cf. [[#References|[a1]]]).

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Introduction to geometry" , Wiley  (1961)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  W.W.R. Ball,   H.S.M. Coxeter,   "Mathematical recreations and essays" , Dover, reprint  (1987)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Stewart,   "Galois theory" , Chapman &amp; Hall  (1973)  pp. Chapt. 5</TD></TR>
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