The following pictures show n points inside a convex region with area 1 so that the area A of the smallest triangle formed by these points is maximized. The smallest area triangles are shown.
Trivial. 3-fold dihedral symmetry.
Trivial. 4-fold dihedral symmetry.
Found by David Cantrell in June 2007. 5-fold dihedral symmetry.
Found by David Cantrell in June 2007. 6-fold dihedral symmetry.
Proved by Zhenbing Zeng and Lu Yang in 1995. 3-fold dihedral symmetry.
Found by David Cantrell in June 2007. 2-fold symmetry.
Found by David Cantrell in June 2007. 2-fold symmetry.
Found by David Cantrell in June 2007. 2-fold dihedral symmetry.
Found by David Cantrell in June 2007. 2-fold symmetry.
Found by David Cantrell in June 2007. 3-fold dihedral symmetry.
Found by David Cantrell in June 2007. No symmetry.
Found by AlphaEvolve in June 2025. No symmetry.
Found by Tej Stead in July 2026. No symmetry.
Found by David Cantrell in June 2007. 4-fold dihedral symmetry.
Found by Tej Stead in July 2026. No symmetry. A = .018238+ Found by Tej Stead in July 2026. 6-fold dihedral symmetry. Found by Tej Stead in July 2026. No symmetry. Found by Tej Stead in July 2026. No symmetry. Found by Tej Stead in July 2026. No symmetry. Found by Tej Stead in July 2026. No symmetry.
Found by Tej Stead in July 2026. No symmetry.
Found by Tej Stead in July 2026. No symmetry.
Found by Tej Stead in July 2026. No symmetry.
Found by Tej Stead in July 2026. No symmetry.
Found by Tej Stead in July 2026. No symmetry.
Found by Tej Stead in July 2026. No symmetry.
3.

A = 1
4.
A = 1/2 = .500
5.
A = (5 – √5) / 10 = .276+
6.
A = 1/6 = .166+
7.
A = 1/9 = .111+
8.
A = .0800+
9.
A = .0640+
10.
A = .0519+
11.
A = 2/47 = .0425+
12.
A = 2/51 = .0392+
13.
A = .03093+
14.
A = .02783+
15.
A = .02456+
16.
A = .0222+
17.
A = .018731+
18.
A = [2 – (3/2 – √(239/3)/6)1/3 – (3/2 + √(239/3)/6)1/3] / 18
19.
A = .014527+
20.
A = .013266+
21.
A = .011650+
22.
A = .010150+
23.
A = .009216+
24.
A = .008512+
25.
A = .007805+
26.
A = .006848+
27.
A = .006200+
28.
A = .005477+