## Projective Geometry - Lecture Notes Chapter 2

### 2.2 The theorems of Desargues and Pappus

**Desargues' Theorem:** In a projective space, two triangles are said to be *perspective from
a point* if the three lines joining corresponding vertices of the triangles meet at a common point called the *center*. Two triangles are said to be *perspective from a line* if the three points of intersection of corresponding lines all lie on a common line, called the *axis*. Desargues' theorem states that if two triangles are perspective from a point then they are perspective from a line. The diagram, called a Desargues Configuration should make this clear.
This theorem is valid in the real projective plane. In other projective planes it may not hold universally, when it does the plane is called a *Desarguesian plane*. The converse of Desargues' theorem is also valid in any Desarguesian plane.

**Theorem 2.2.1:** Let V be a vector space of dimension d + 1 over a division ring F. Then the theorem of Desargues holds in **P**(V).

**Pappus' Theorem:** If points A,B and C are on one line and A', B' and C' are on another
line 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*. Every
pappian plane is also Desarguesian.

**Theorem 2.2.2:** Let V be a vector space over the division ring F. Then the theorem of Pappus holds in **P**(V) if and only if F is commutative (i.e., F is a field).

**Theorem 2.2.3:** (*Hessenberg's Theorem*) Let **P** be an arbitrary projective space. If the theorem of Pappus holds in **P** then the theorem of Desargues also holds in **P**.