Classical Geometry: Euclidean, Transformational, Inversive, and Projective

CHAPTER 16

INTRODUCTION TO PROJECTIVE GEOMETRY

16.1 Straightedge Constructions

We saw earlier that a compass alone is as “powerful” as a compass combined with a straightedge. We begin this section by indicating why a straightedge alone is not as powerful as a straightedge and compass or a compass alone. There are only a few admissible operations that can be done with a straightedge by itself.

Admissible Straightedge Operations

1. Draw an arbitrary line.

2. Draw a line through a given or previously constructed point.

3. Draw a line through two given or previously constructed points.

4. Construct a point as the intersection of two different lines.

A straightedge construction is a finite sequence of the above operations.

We will give informal proofs that certain well-known constructions with straightedge and compass are not possible with a straightedge alone.

One of the standard straightedge and compass constructions is bisecting a given line segment.

Theorem 16.1.1. Using only a straightedge, we cannot construct the midpoint of a given segment.

Proof. The idea behind the proof is that a straightedge construction is projectively invariant. Here we give an intuitive justification of the theorem.

Suppose that there is a finite sequence of the possible straightedge operations that yield the midpoint of a segment AB. In other words, there is a sequence of instructions that, when followed, produces the midpoint of AB. For example, the first few instructions might be:

(1) Draw a line ...