## Lecture Notes 10: Pappus' Theorem

We will again consider a "theorem" of Euclidean geometry in the projective plane, which may or may not be valid (i.e., a property which may or may not hold).
**Pappus' Theorem:** If points A,B and C are on one line and A', B' and C' are on another
line (but none at the point of intersection of these lines) then the points of intersection of the lines AC' and CA', AB' and BA', and BC' and CB' lie on a common line called the *Pappus line* of the configuration.

This theorem is valid in the real projective plane, but may not be valid universally in other projective planes. When it is universally valid, the plane is called a *Pappian plane*.

When the Pappus line is the line at infinity we obtain the following affine version of the Pappus theorem:

**Pappus' Theorem (affine version)**: Let O be the intersection of two lines, *l* and *m*, in an affine plane. If points P, Q and R are on *l* and S, T and U are on *m*, then if the line QS is parallel to RT, and PT is parallel to QU, then the line PS is parallel to PU.

**Theorem**: Pappus' theorem is true in a Desarguesian affine plane if and only if multiplication on the points of its lines is commutative.

We now return to properties of multiplication of points on a line in a Desarguesian affine plane.

**Theorem**: For points A,C and E on line *l*, (A+C)E = AE + CE.

**Theorem**: For points C and E on *l*, E(I+C) = E + EC.

**Corollary**: For points A,C and E on *l*, E(A+C) = EA + EC.

We have at this point shown that with the addition and multiplication that we have defined for points on a line in a Desarguesian affine plane, we have constructed a division ring. Furthermore, if this affine plane satisfies the Pappus theorem, then the division ring is a field.

To clarify the relationship between the Desarguesian and Pappian properties, we mention the following results.

**Theorem:** (*Hessenberg's Theorem*) Let **P** be an arbitrary projective plane. If the theorem of Pappus holds in **P** then the theorem of Desargues also holds in **P**.

We also know that by Wedderburn's theorem, every finite division ring is a field, and hence has a commutative multiplication. Therefore, we can conclude that every finite Desarguesian plane is Pappian. There are examples of infinite Desarguesian planes which are not Pappian.