Square, n = 3 proven

A = 12
= 0.500000000000000000000000000000000000000000000…
A unique minimal triangle.

The family of optima

The optimum is not unique: with the two lower corners fixed, the third point may sit anywhere on the top edge — the triangle keeps base 1 and height 1, so its area is exactly ½ for every x ∈ [0, 1]. One slice of the full family of area-½ triangles.

Symmetry

Mirror symmetric (group D1, order 2).

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

Minimal triangles

1 triangle tied (within relative 10−9) at the minimal area:

countside lengthstriangles (point indices)
1 1.0000 · 1.0000 · 1.4142 (0,1,2)

Provenance

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