Square, n = 6 proven

A =

\frac{1}{8}

= 0.125000000000000000000000000000000000000000000…

6 minimal triangles in 2 congruence classes, colored by class.

Symmetry

2-fold dihedral symmetry (group D2, order 4). (Friedman's page says: vertically and horizontally symmetric.)

6 triangles tied (within relative 10⁻⁹) at the minimal area, in 2 congruence classes:

	count	side lengths	triangles (point indices)
	4	`0.5000 · 0.7071 · 1.1180`	(0,1,2) (0,3,4) (1,2,5) (3,4,5)
	2	`0.5000 · 0.5000 · 0.7071`	(0,1,4) (2,3,5)

Proved optimal by A. Dress, L. Yang, and Z. B. Zeng, 1995.
One of an infinite family of solutions.
Coordinates from spiralulam/heilbronn (MIT): spiralulam/heilbronn config_n06.json — Dress, Yang, Zeng (1995).
Verified in exact arithmetic: all 20 triples enumerated, 6 tied at the minimum.
Friedman's page lists: A = 1/8 = .125.