Convex, n = 8 proven

A = the unique real root of

2060 x^{5} - 2332 x^{4} + 1064 x^{3} - 240 x^{2} + 26 x - 1

= 0.0800001393294664385749398892640329663025550560…

optimality machine-checked in Lean 4

10 minimal triangles in 6 congruence classes, colored by class.

Symmetry

Mirror symmetric (group D1, order 2).

8 points in 5 orbits (2 + 2 + 2 + 1 + 1) — hover a point to see its orbit.

10 triangles tied (within relative 10⁻⁹) at the minimal area, in 6 congruence classes:

Found by David Cantrell, June 2007.
Proved optimal by Tej Stead, September 2026.
The optimum is the unique real root of 2060x⁵ − 2332x⁴ + 1064x³ − 240x² + 26x − 1; the optimizer — a reflection-symmetric heptagon with one interior point — is unique up to relabeling and invertible affine maps. Machine-checked in Lean 4 and registered in the Palomar registry (PALOMAR-2026-09-02-000012).
Cantrell's 2007 coordinates (shown) agree with the proven optimum to 1.4 × 10⁻¹⁶.
Coordinates by an external contributor, re-verified here: David Cantrell's Mathematica notebook UltimateHeilbronn.nb (2007), received September 2026 — Original author's arrangement, previously shown only as a reconstruction; given exactly in the notebook..
Verified in exact arithmetic: all 56 triples enumerated, 10 tied at the minimum.

2026-09-02 marked proven: the value is the unique real root of 2060x⁵ − 2332x⁴ + 1064x³ − 240x² + 26x − 1, machine-checked in Lean 4 (Tej Stead, September 2026); Cantrell's 2007 configuration matches the optimum to 1.4 × 10⁻¹⁶
2026-09-03 proof registered: Palomar registry entry PALOMAR-2026-09-02-000012 v1