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
  1. preserves the ratio of parallel segments, and
  2. preserves the signed ratio of collinear points.

proof of the affine ratios theorem

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.

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