An analog of the Furstenberg-Katznelson-Weiss theorem on triangles in   sets of positive density in finite field geometries
  We prove that if the cardinality of a subset of the 2-dimensional vector space over a finite field with $q$ elements is $\ge \rho q^2$, with $\frac{1}{\sqrt{q}}<<\rho \leq 1$, then it contains an isometric copy of $\ge c\rho q^3$ triangles. 
David Covert, Derrick Hart, Alex Iosevich and Ignacio Uriarte-Tuero