For integer $$A$$, a grid of $$n$$ points has roughly $$n^2\sigma(A)/A$$ area-$$A$$ triangles, where $$\sigma$$ is the sum of divisors; see Erdős & Purdy 1971 (users.renyi.hu/~p_erdos/1971-2) who used non-square grids and factorial $$A$$ to find points with $$\Omega(n\log\log n)$$ unit-area triangles. So how big is $$\sigma(A)/A$$? It depends on the Riemann hypothesis! If RH is true, at most $$e^\gamma\log\log A$$ for $$A>5040$$. If not, slightly larger infinitely often. See en.wikipedia.org/wiki/Divisor_

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