Your task is this:
Find all arrangements of four distinct points in the plane such that only two distances occur between them.
When you have $4$ distinct points, you can measure the distance between any two of them. There are $6$ pairs of points, so in general you can measure $6$ different distances. There are however some configurations in which several of those distances are the same. While they can't all be equidistant (at least not in the flat plane), it is possible for only $2$ distances to occur between the points.
It is easy to find one or two of these arrangements, but a little tricky to find them all, as it is very easy to overlook one. When you answer this question, please try to include some explanation for why you think you got them all.
Source: I saw this question on Colin Wright's blog, who heard it from Peter Winkler.