  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 Koml\'os, Pintz and
Szemer\'edi 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.
