## Lecture Notes 9: Multiplication on a Line

Given a line *l* in a Desarguesian affine plane and two points O and I on *l*, we define a multiplication of points A and C on *l* by:
- Pick a point B not on
*l*.
- Let
*m* be the line through A parallel (or equal) to the line through I and B.
- Let D be the intersection of
*m* and the line through O and B.
- Let
*k* be the line through D parallel (or equal) to the line through B and C.
- Define AC to be the intersection of
*k* and *l*.

**Theorem**: For any point A on *l*, AI = IA = A.
**Theorem**: For any point A on *l*, AO = OA = O.

**Theorem**: If AO, there is a point (A^{-1}) on *l* such that A(A^{-1}) =(A^{-1})A = I.

**Theorem**: The definition of the point AC is independent of the choice of B.

**Theorem**: For any points A, C and E on *l*, (AC)E = A(CE).

The above theorems show that the operation of multiplication on the points of a line turns that set into a group (not necessarily abelian).