Angle trisection

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The problem of trisecting the angle is a classic problem of compass and straightedge constructions of ancient Greek mathematics.

Problem: construct an angle one-third a given arbitrary angle, given only two tools:

With such tools, it is generally impossible, as shown by Pierre Wantzel (1837). This requires taking a cube root, impossible with the given tools. The fact that there is no way to trisect an angle in general with just a compass and a straightedge does not mean that it is impossible to trisect all angles so.

Contents

Perspective and relationship to other problems

Using only an unmarked straightedge and a compass, Greek mathematicians found means to divide a line into an arbitrary set of equal segments, to draw parallel lines, to bisect angles, to construct many polygons, and to construct squares of equal or twice the area of a given polygon.

Three problems proved elusive, specifically:

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