Section 2.2 Constructions
¶Classic constructions in Euclidean geometry are made using just a straightedge (a ruler without markings) and a compass (a tool with two “legs” for drawing circles of arbitrary radius). Modern axioms for Euclidean geometry also include postulates for measuring lengths and angles; here, the only question is whether geometric objects such as line segments are congruent, that is, whether they have the same length even though we don't know what that length is.
What can we do with these tools? The straightedge is used to draw the line (or line segment) between any two points. The compass can be used to draw circles (or circular arcs) with center at a given point, passing through any other point. But it can also be used to duplicate lengths. How? By placing the two legs of the compass on the endpoints of a line segment, thus fixing a scale, then moving the compass to a new center point and drawing a circle. The radius of this circle is then the distance between the two original points.
Most drawing programs, including GeoGebra, have a separate tool for this latter, fundamental use of a compass.
One of the most basic constructions is to make a right angle. This construction can be accomplished using a straightedge and compass as shown in Figure 2.2.1. Given the initial line segment shown in the figure, first extend it to twice its length (using a compass to draw the small circle shown in the figure), then bisect the resulting segment (using two more circles), then connect the dots.
Two other angles are easy to construct, as can already be seen from Figure 2.2.1. Complete the triangle in this figure. The remaining angles are \(30^\circ\) and \(60^\circ\text{.}\) Can you see why? Using essentially the same construction, an equilateral triangle can be constructed as shown in Figure 2.2.2.
Are there any other angles we can construct? Well, we can bisect any angle by connecting four points on the legs in an “X” pattern, as shown in Figure 2.2.3. Since angles can be added, we can in principle approximate any desired angle by successive bisections and/or sums. But that approach is not very elegant. It can in fact be shown that it is not possible to trisect an angle using straightedge and compass alone.