The Proof of The Fundamental Theorem of Affine Geometry

The Fundamental Theorem of Affine Geometry
If L =(A,B,C) and L' = (A',B',C') are lists of non-collinear points of R2,
then there is a unique element of A(2) mapping L to L'.

This follows quickly from

The (O,X,Y) Theorem
If L =(P,Q,R) is a list of non-collinear points of R2, then
there is a unique element of A(2) mapping (O,X,Y) to L,
where O=(0,0), X = (1,0), Y = (0,1).

Proof of the (O,X,Y) Theorem
Let P, Q and R have position vectors p, q and r respectively.
As P, Q and R are non-collinear, (q - p) and (r - p) are
non-parallel, and hence linearly independent.
Let t be the affine transfromation t(x) = Ax + b.
Then t maps O, with position vector 0, to b.
Observe that, if A has columns c and d, then
it maps X to c + b and Y to d + b.
Thus it maps (O,X,Y) to (P,Q,R) if and only if
b = p,
c + b = q, i.e. c = q - p, and
d + b = r, i.e. d = r - p
Since these values of c and d are linearly independent,
they are the columns of an invertible matrix A.
Thus, there is a unique affine transformation t of the required type.

Proof of the Fundamental Theorem
By the (O,X,Y) Theorem, there exist elements r, s of A(2)
such that r maps (O,X,Y) to L and s maps (O,X,Y) to L'.
Then t = sor-1 maps L to L'.

Suppose that u also maps L to L'. Then uor maps (O,X,Y) to L'.
By the uniqueness clause of the Theorem, only s maps (O,X,Y) to L' .
Thus uor = s, so u = sor-1 = t, i.e. t is unique.

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