Show that every mapping which preserves between-ness is a collineation I had a geometry class which was proctored using the Moore method, where the questions were given but not the answers, and the students were the source of all answers in the class.  One of the early questions which we never solved is listed in the title.
In this case, use any reasonable definition of "between-ness".  I believe the definition we used was "$B$ is between $A$ and $C$ if and only if $|AB|+|BC|=|AC|$".  A collineation is a mapping where every line is mapped to a line.  A mapping is a function that operates over the set of points within the given space and returns points in the given space.
When we were studying this question, we managed to get to the point that a between-ness mapping must map lines to line segments.  In particular, we had managed to show that for every $A-B-C$ (read: "$B$ is between $A$ and $C$") in the mapped space by applying between-ness preserving mapping $m$, we could guarantee that pre-images $P_A, P_C$ such that $m(P_A)=A,m(P_C)=C$ implies the existence of pre-image $P_B$ such that $P_A-P_B-P_C$ and $m(P_B)=B$.  I have never seen any full proof of the title statement.
I would like to read any hints that are known for solving this question.  Feel free to completely solve it, but please hide the full solution in such a way that I can start with your hint(s) and have an opportunity to finish the solution for myself.
 A: 
A collineation is a mapping where each pair of parallel lines is mapped to a pair of parallel lines.

In my (projective) vocabulary, a collineation is a mapping which maps collinear point triples to collinear point triples. Preservation of parallelity is not implied. The only problem with transformations which don't preserve parallelity is that they might map points in the plane to infinity and vice versa, so you'd need a projective framework to properly express these.
In the common Euclidean (non-projective) plane, if three points are collinear, then one of them is between the other two. If that between-ness is preserved, then the equation you stated holds for the image points as well. But the only way for this equation to hold is if the image points are again collinear. Thus your map must bee a collineation, in my sense as written above.
If you require preservation of parallelity, then I guess the key to that is in what you consider a mapping. Every collineation in the real projective plane is a projective transformation. (The same isn't true for projective planes over fields with non-trivial automorphisms, e.g. over the complex numbers.) A projective transformation which will not send finite points to infinity must fix the line at infinity as a whole. And any projective transformation which fixes the line at infinity is an affine transformation, and as such will map parallel lines to parallel lines. So if the domain of the mapping is the set of all finite points, and its codomain does not include points at infinity, then preservation of parallel lines is guaranteed.
A: In an affine plane, there is a unique line through every two points. You can cover any of the lines with line segments. What a line segment-preserving map is then doing is sending all the points on a line to points on another line, and that line can be considered the image of line itself under the mapping.
