Example

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 $AB$, we would like to construct the midpoint $M$ of that segment. Then we have bisected the segment, since $AM = MB$.

Figure 1: Bisecting a segment--steps 1 and 2

Figure 1 shows the first two steps. A compass is used with center at $A$ to draw a circle passing through $B$, and in the same way, another circle is drawn centered at $B$ and passing through point A.

Figure 2: Bisecting a segment--steps 3 and 4

In Figure 2, two new points are determined--the intersections of the two circles that were drawn in step 1. Label these two points $C$ and $D$.

Figure 3: Bisecting a segment--steps 5 and 6

The final two steps are shown in Figure 3. The straightedge is used to connect points $C$ and $D$ with a new line, and finally, the intersection $M$ of that line with the original segment $AB$ is determined. $M$ is the required midpoint. (The other segments in the figure, such as $CA$, $CB$, $DA$, and $DB$ are not part of the construction, but we will use them below.)

It may seem obvious from the figure that $M$ 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 $CA$, $CB$, $DA$, and $DB$ are equal, since they are radii of two circles with radius $AB$. Therefore the quadrilateral $ACBD$ is a parallelogram, and we know that the diagonals of a parallelogram bisect each other. There are dozens of other easy proofs.