Triangle, n = 6 proven

A =

\frac{1}{8}

= 0.125000000000000000000000000000000000000000000…

6 minimal triangles in 5 congruence classes, colored by class.

The family of optima

One 1-parameter slice of the optimal set: point 2 slides down the left edge for y ∈ [1/4, 3/8] (right-frame coordinates; the stored member is the top endpoint). Its motion is parallel to the segment through points 0 and 5, so the tied triangle (0,2,5) keeps its area; the released triangle (2,3,5) grows. Every member achieves exactly A = ⅛.

Symmetry

Not symmetric (group C1, order 1).

Minimal triangles

6 triangles tied (within relative 10⁻⁹) at the minimal area, in 5 congruence classes:

count	side lengths	triangles (point indices)
2	`0.6580 · 0.7598 · 1.3698`	(0,4,5) (1,4,5)
1	`0.5026 · 0.5026 · 0.7598`	(0,2,5)
1	`0.3799 · 0.6580 · 0.7598`	(1,3,5)
1	`0.3799 · 0.7598 · 1.0052`	(0,1,3)
1	`0.5026 · 0.6580 · 1.0576`	(2,3,5)

Provenance

Found by L. Yang, J. Z. Zhang, and Z. B. Zeng, 1991.
Proved optimal by L. Yang, J. Z. Zhang, and Z. B. Zeng, 1994.
One of an infinite family of solutions.
Coordinates exact, as published in: arXiv:2607.15021 — one member of the optimal family.
Verified in exact arithmetic: all 20 triples enumerated, 6 tied at the minimum.

Downloads

points.txt coordinates, tab-separated
points.csv coordinates, CSV
points.json full record: value, provenance, verification
figure.svg this figure
family.json the optimal family: exact endpoints, 30-decimal sample members

References

L. Yang, J. Z. Zhang, Z. B. Zeng, A conjecture on the first several Heilbronn numbers and a computation, Chinese Ann. Math. Ser. A 13 (1992) 503–515