Question 1107832: I'm a student of MSciEd secondary math.. I just wanna ask if you could answer the problem set that my instructor in mathed 235 haave given us.. the question is
Determine whether or not there exists a finite set M of points in space not lying in the same plane such that, for any two points A and B of M, one can select two other points C and D of M so that the lines AB and CD are parallel and not coincident
Answer by ikleyn(52914)   (Show Source):
You can put this solution on YOUR website! .
Such a set does not exist.
Proof
Let assume for a minute that the finite set M of points with such properties does exist.
Then you can select the pair of points (A,B) such that the distance d(A,B) between the points is maximal among all pairs of points from M.
So, you selected such a pair.
According to properties of M, there is ANOTHER, distinct pair of points (C,D) such that the segments AB and CD are parallel.
Then the segments AB and CD lie in one plane, so the quadrilateral ABCD is a plane quadrilateral and all its vertices lie in the same plane.
Thus the quadrilateral ABCD (with the verices listed in the correct order) is either trapezoid or parallelogram.
In any case, at least one of the two diagonals of such a qudrilateral is longer than its side AB.
It is just a contradiction, since we assumed that d(A,B) was the longest distance.
The contradiction PROVES the statement.
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