proof of the affine ratios theorem

The Affine Ratios Theorem
An affine transformation t
  1. preserves the ratio of parallel segments, and
  2. preserves the signed ratio of collinear points.

(1) Suppose that PQ and RS are parallel segemnts, and that PQ/RS = k.
Say t maps P,Q,R,S to P',Q',R',S', respectively. As usual, we will write
m for the position vector of the point M. Finally, as t is affine, it has
the form t(x) = Ax+b.
Then P'Q' = q' - p' = (Aq+b)-(Ap+b) = A(q-p), = APQ
and, similarly, R'S' = A(s-r) = A(RS).
Since PQ = kRS, we have P'Q' = kR'S', as required.

(2) is immediate, since signed ratio is a special case of the ratio of
parallel segments.

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