We will investigate point sets in the plane with many special triples. There are three types of special triangles (triples) we will focus on:

- collinear triples

- triples similar to a given triangle

- triples with unit circumcircle

All three cases lead us to hard problems in discrete geometry. The general conjecture is that if a finite planar point set has many special triples then it has a large, structured subset which is rigid in a properly defined way.