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Angle Trisection by Construction of a Well-Defined Locus and a Simple Arc

Brian R. Stonebridge , Angle Trisection by Construction of a Well-Defined Locus and a Simple Arc . CSTR-96-004, Department of Computer Science, University of Bristol. June 1996. No electronic version available.

Abstract

A well-known problem of Euclidean geometry is to trisect a given angle by using the standard operations of straight edge and a pair of compasses. Whilst this has been regarded as impossible since as long ago as the Greeks, proofs of the result rely upon more modern mathematics, such as the fact that the operations provided are rational but the solution requires surds, that is, roots which are irrational. However, with the graphics ability of modern packages, one can give the impression of performing this task and, indeed, can actually do so to the accuracy provided by the resolution of the screen. At the same time, some interesting mathematical properties, which might easily otherwise go unnoticed, are brought to light. The methods stem from the construction of another angle comprising three equal angles, forming the trisectors of that angle. By using Morley's trisector theorem and properties of the angles of the diagram, the point of intersection of two trisectors is obtained as the point at which a well-defined locus is incident on a simple arc of a circle. This prescribes a side of the equilateral triangle and hence the unique trisectors of the given angle.

Bibtex entry.

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