Triangle, n = 5 proven

A =

3 - 2 \sqrt{2}

= 0.171572875253809902396622551580603842860656249…

4 minimal triangles in 3 congruence classes, colored by class.

The family of optima

One 1-parameter slice of the optimal set: point 1 slides along the hypotenuse, (1 − x, x) for x ∈ [2 − √2, 1/√2] (right-frame coordinates; the stored member is the x = 2 − √2 endpoint). Its motion is parallel to the segment through points 3 and 4, so the tied triangle (1,3,4) keeps its area; the released triangle (1,2,4) grows. Every member achieves exactly A = 3 − 2√2.

Symmetry

Mirror symmetric (group D1, order 2).

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

Minimal triangles

4 triangles tied (within relative 10⁻⁹) at the minimal area, in 3 congruence classes:

count	side lengths	triangles (point indices)
2	`0.6295 · 0.7927 · 1.3225`	(0,3,4) (1,2,4)
1	`0.6295 · 0.6295 · 0.6295`	(1,3,4)
1	`0.7927 · 0.7927 · 1.5197`	(0,2,4)

Provenance

Proved optimal by Royce Peng, 1989.
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 10 triples enumerated, 4 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