A harder version of Erdős’ famous distinct-distances problem asks about the structure of point sets in the plane that have few distances. In particular, Erdős asked if such near-optimal sets have a ‘lattice structure’ or, given this appears very hard, if a polynomial proportion of the points lie on a line. In joint work with Sam Mansfield we consider sets with few congruent triangles, showing such sets contain either a polynomially-rich line or a positive proportion of the set on a circle. There will also be some discussion of energy results in such a setting. Our methods use a Balog-Szemerédi-Gowers type result over the Euclidian group along with classical results from additive combinatorics.