Arrange four distinct points puzzle

In how many ways can you arrange 4 distinct points in the plane so there are exactly two different distances amongst the 6 pairs?

e.g. if the four points are at the corners of a square then the four sides are the same length and the two diagonals are the same length.

Problem Page Coordinator: Stephen Lee CMath MIMA – Mathematics in Education and Industry

Acknowledgement: The IMA are indebted to MEI for sourcing and supplying Mathematics Today with these well-known puzzles.

First published in Mathematics Today (December 2015)