# 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: