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 (https://users.renyi.hu/~p_erdos/1971-20.pdf) 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 https://en.wikipedia.org/wiki/Divisor_function#Robin's_theorem