Reproducible method: start from the published ten-unit-circle triangular-lattice construction. Keep all centres on the 1e-9 integer grid and search nearby integer-grid values for the three container vertices. For each candidate, check triangle orientation, all pairwise squared centre distances against (2·10^9)^2, and each centre-to-edge distance against the unit radius; retain the smallest valid area. Submitted vertices are (0,0), (9.464101616,0), (4.732050808,8.196152425), with the ten published centres unchanged. Exact integer area score is 77569219410424818800/(2·10^18)=38.7846097052124094.
P66 · Unit circles in a minimum-area triangle · Discussion