The best introduction to classical geometric construction is to show an example.
In Section 3, the ``official'' rules for geometric construction will be presented, but we'll begin with just an intuitive idea of what can be done with a straightedge and compass.
What we would like to do is to begin with a line segment, and using only a straightedge and compass, divide that line into two equal segments. In other words, if we are given a line segment , we would like to construct the midpoint of that segment. Then we have bisected the segment, since .
Figure 1 shows the first two steps. A compass is used with center at to draw a circle passing through , and in the same way, another circle is drawn centered at and passing through point A.
In Figure 2, two new points are determined--the intersections of the two circles that were drawn in step 1. Label these two points and .
The final two steps are shown in Figure 3. The straightedge is used to connect points and with a new line, and finally, the intersection of that line with the original segment is determined. is the required midpoint. (The other segments in the figure, such as , , , and are not part of the construction, but we will use them below.)
It may seem obvious from the figure that is the midpoint, but part of any construction is a proof that the required point has been found. In this case it is very easy to do--notice that all the lengths , , , and are equal, since they are radii of two circles with radius . Therefore the quadrilateral is a parallelogram, and we know that the diagonals of a parallelogram bisect each other. There are dozens of other easy proofs.