# Geometry, equating two sets

If $A,B,C$, and $D$ are 4 different collinear points, then there exists points $X,Y,Z,W$ such that $\{A,B,C,D\}=\{X,Y,Z,W\}$ and $XYZW$ is true.

I need help understanding what this is actually trying to say and where the proof is headed.

My attempt: Since $A,B,C$ are points and $B,C,D$ are different points and are collinear, then a couple things are true: $ABC$, $BCA$, or $CAB$ as well as $BCD$,$CDB$, or $DBC$. But I don't see how this helps me or if it even does. This problem is approached using some basic axioms such as if $ABC$ and $BCD$ is true, then $ABCD$ is true, etc.

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Are you using the word "true" as a synonym for "collinear?" Is there any reason you can't set $X=A,Y=B,Z=C,W=D?$ – Kevin Carlson Sep 6 '12 at 5:12
Sorry, I'm meaning true in the sense that it is a fact derived from axioms. See that's also a thought that I had, but I'm not sure what the theorem is even stating. – user23793 Sep 6 '12 at 5:15
It looks as if you are trying to prove a betweenness result. Is that so? – André Nicolas Sep 6 '12 at 5:15
I think I need to find a way to make ABCD true and let Kevin's idea play a role. – user23793 Sep 6 '12 at 5:17
I think when you write "XYZW is true" what you mean is "Y is between X and Z, and Z is between Y and W." If that's right, could you edit that information into your question? And maybe let us know exactly which axioms you are allowed to use? – Gerry Myerson Sep 6 '12 at 7:24