A classical problem in geometric combinatorics is to determine how often a single distance may repeat among $n$ points in the plane. A related problem that has received much attention is how often a given triangle can repeat among $n$ points in the plane. We shall report on some partial progress on the second problem using a continuous analog, proved via a Sobolev estimate for a bilinear averaging operator and a conversion mechanism that allows us to deduce a discrete result from a sufficiently robust continuous analog.