# 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)