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.