The approach of

affine geometry, as opposed to the approach of

analytic geometry, uses vector methods to reveal geometric facts in a simple manner. This approach dates back to

Felix Klein (1849-1925) and

Sophus Lie (1842-1899).
To quote my source book: "It gives concrete examples leading to an appreciation of the theory of

groups", hence a knowledge of

Abstract algebra can be useful.

Euclidean geometry can be seen as the geometry associated to the group of

isometries.
Here are some basic definitions to get a feel of

affine geometry.

A point is defined by two values, (

*a*_{1},

*a*_{2}), its

Cartesian coordinates.
A point is also considered to be a vector from the origion to the location of the point. Two vectors are equal if their coordinate values are equal.

If

*A* = (

*a*_{1},

*a*_{2}) and

*B* = (

*b*_{1},

*b*_{2}),
the addition of vectors is defined as

*A* + *B* = (*a*_{1} + *b*_{1}, *a*_{2} + *b*_{2}).

Multiplication by a scalar is defined as:

*rA* = (*a*_{1}, *ra*_{2}).

When using abstract

vectors,

division is not defined.
However, what is traditionally the division symbol is used. For example, if a point

*P* lies on line

*l*_{AB} (line passing both

*A* and

*B*), distinct from

*B*, then (

*P* -

*A*)/(

*P* -

*B*) =

*b/a*
is an alternative to writing (

*P* -

*A*) = (

*b/a*)(

*P* -

*B*) where

*a* and

*b* are

scalers.

Point

*P* is on the line

*l*_{AB} iff
there exists a scalar

*s*:

*P* = *A* + *s*(*B* - *A*)

This also means

*P* = *rA* + *sB* (where *r* + *s* = 1)

The

midpoint *M* can be written as

*M* = (1/2) (

*A* +

*B*), where

*M* is defined as a

midpoint of

*A* and

*B* iff *A* -

*M* =

*M* -

*B*.

In addition to these operations,

dot product is used to define

orthogonality, and the

projection function is used to define reflection

isometries.

Theorems such as the

theorem of Thales,

theorem of Menelaus,

theorem of Ceva, the

nine point circle theorem, etc.
can be proven taking the

affine geometry approach.

Source:
"Vectors and Transformations in Plane Geometry" by Philippe Tondeur, Publish or Perish, Inc. 1993