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, (a1, a2), 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 = (a1, a2) and B = (b1, b2), the addition of vectors is defined as
A + B = (a1 + b1, a2 + b2).
Multiplication by a scalar is defined as:
rA = (a1, ra2).
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 lAB (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 lAB 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

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