Trisection of an angle

The problem of dividing an angle $\phi$ 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 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 construcible numbers. 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

[1]	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)

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 [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. [a1]).

References

[a1]	H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)
[a2]	W.W.R. Ball, H.S.M. Coxeter, "Mathematical recreations and essays" , Dover, reprint (1987)
[a3]	I. Stewart, "Galois theory" , Chapman & Hall (1973) pp. Chapt. 5