For sufficiently large $n$, we show that in every configuration of $n$ points chosen inside the unit square there exists a triangle of area less than $n^{-8/7-1/2000}$. This improves upon a result of Komlós, Pintz and Szemerédi from 1982. Our approach establishes new connections between the Heilbronn triangle problem and various themes in incidence geometry and projection theory which are closely related to the discretized sum-product phenomenon.
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