A triangle T′ is ε-similar to another triangle T if their angles pairwise differ by at most ε. Given a triangle T, ε>0 and n∈N, Bárány and Füredi asked to determine the maximum number of triangles h(n,T,ε) being ε-similar to T in a planar point set of size n. We show that for almost all triangles T there exists ε=ε(T)>0 such that h(n,T,ε)=n3/24(1+o(1)). Exploring connections to hypergraph Turán problems, we use flag algebras and stability techniques for the proof.