## The Theorem of Pappus

**Interactive Course on Projective Geometry**

In the geometric constructions that follow, we will denote points by
uppercase letters and lines by lowercase letters. The symbol
denotes the line passing through the points and , while
denotes the point of intersection of the lines and
.

**Theorem 0.1** (Pappus)
Draw two lines on the projective plane and three points on each line.
Denote the points on the first line by

, and the points
on the second line by

. Draw the lines that join
points denoted by different letters (i.e., we do not draw the lines

, or

). The points

,
and

are collinear.

The proof is given in the following two exercises, the first of which is to
prove the affine version of the theorem.

**Exercise 0.1** (10)
Draw two lines on the plane and draw points

on the
first line, and points

on the second line such that

is parallel to

, and

is parallel to

. Show
that

is necessarily parallel to

.

Now comes the reduction of the projective result to the affine
result.

**Exercise 0.2** (10)
Let

be any projective transformation that sends the points

and

to infinity. Apply

to the
points and lines in the configuration of theorem

0.1 and verify
that the new configuration is like the one of the previous exercise.
Use this to prove Pappus' theorem.