affine properties

From Theorem A0, we see that collinearity and parallelism are affine properties.
We know that length and angle are not invariant under affine transformations.
Here, we establish a result about parallel segments which leads to many useful
affine concepts.

Suppose that AB and CD are parallel segments. Then, in terms of vectors, then
we have AB = kCD, for some real k. We refer to this as the ratio of the segments
and denote it by AB/CD.

Now suppose that P, Q and R are collinear points. Then PQ and QR are parallel
segments, so that their ratio k is defined. In this case, we call k the signed ratio
of the points P, Q, R (in order), and denote it by PQ/QR.

 The Affine Ratios Theorem An affine transformation t preserves the ratio of parallel segments, and preserves the signed ratio of collinear points. Observe that point Q lies between points P and R if and only if the signed ratio is positive. Thus betweenness is an affine concept. Similarly, Q is the mid-point of PR if and only if PQ/QR =1, so that affine transformations map mid-points to mid-points. A median of a triangle is a segment joining one vertex to the mid-point of the opposite side. Hence the above observation about mid-points shows that affine transformations map medians to medians. Thus mid-point and median are affine concepts.