A distance between a pair of points in an affine space is invariant under translation, rotation and reflection.
An angle in a triangle whose corners are tree points is also invariant under scaling.
Which function of four points is invariant under, possibly, affine transformations?
What is the general definition of such invariants?
The simplest affine invariant seems to be the ratio of the lengths of two line segments in the same direction (lying on either a single line or two parallel lines).