The Fundamental Theorem of Affine Geometry If L =(A,B,C) and L' = (A',B',C') are lists of noncollinear points of R^{2}, then there is a unique element of A(2) mapping L to L'. This follows quickly from
The (O,X,Y) Theorem


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 noncollinear, (q  p) and (r  p) are nonparallel, 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
Suppose that u also maps L to L'. Then uor maps (O,X,Y) to L'.
