Section 2.3 Constructing a Compass
¶A straightedge is simply a tool for drawing lines. The incidence postulates ensure that two points determine a unique line, so a straightedge is nothing other than a tool for producing that line. Similarly, the ruler and angle postulates ensure that circles exist, and we can easily imagine simple tools for drawing circles, without necessarily knowing the value of the radius. In fact, one of Euclid's original postulates asserts precisely that circles of any size can be drawn. Euclid then showed how to construct a compass, which is a tool for drawing a circle of a given size, namely the length of a given line segment.
As shown in Figure 2.3.1, the goal is to duplicate line segment \(AB\) at point \(C\text{.}\) Start by constructing an equilateral triangle \(BCP\text{,}\) using the construction in Section 2.2. Also construct a circle with radius \(AB\text{,}\) and extend line segment \(PB\) so that it meets this circle at \(Q\text{.}\) The distance from \(P\) to \(Q\) is therefore the sum of the distances from \(A\) to \(B\) and \(B\) to \(C\text{,}\) and the latter is of course equal to the distance from \(P\) to \(C\text{.}\) Drawing the circle at P whose radius is this combined distance, and extending \(PC\) until it meets this circle at \(D\text{,}\) results in \(CD\) having the desired length.