During the development of early mathematics, it was discovered that through the use of nothing more than a compass and a straight edge, many geometric figures could be exactly built. With the development of geometric theroems, it became possible to prove that these figures were what they appearred to be(The development of non-Euclidean geometry forced a rethinking of constructions, but for Euclidiean space, the system still holds). A simple example is building a line perpendicular to a segment and passing through the mid-point of the segment.

Given a segment, draw a circle centered at one of the endpoints and containing the other. Then, draw a similar circle of the same size centered at the other endpoint and containing the first. Then draw a line through the two intersection point of the circles. This line is then perpendicular to segment AB, and the intersection of the line and segment is the mid-point of the segment. Thus, a perpendicular bisector.