Ruler-and-compass constructions

Abstract

Abstract In the Greek tradition of geometry there has been much interest in determining which geometrical constructions can be performed using only two very simple devices. This question is asked in an idealized context where constructions can be performed with complete accuracy. The first permitted device is a ruler or straight edge; this makes it possible to draw the line joining distinct points A and B in the plane and to extend the line as far as necessary in either direction. The ruler has no markings on it and it cannot be used for measuring lengths. The second device is a pair of compasses; this makes it possible to draw the circle with centre at given point A passing through a second point B. It was known to the ancient Greek geometers that it is possible, using only these devices in combination, to perform such constructions as bisecting a given angle or dropping a perpendicular from a point to a line (details below).