# 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