How many ways can four points be arranged in a plane so that the six distances between pairs of points take on only two different values?
If the points do not need to be distinct, then there are 3 or 4 more cases:
a) Three points at one location, and one point elsewhere
b) Two points at one location and two at a different location
c) An equilateral triangle with two points at one vertex.
We can get rid of these trivial solutions if the points must be distinct, or if the distances need to be > 0.
I don't think that
d) All 4 points co-located,
satisfies the problem as stated, since all six pairwise distances are
equal. In other words, we don't have two different values!
And 3 or 4 trivial solutions
