The Heilbronn Problem for Triangles

The following pictures show n points inside a triangle with unit area so that the area A of the smallest triangle formed by these points is maximized. The smallest area triangles are shown.

3.
A = 1

Trivial.

Completely symmetric.

4.
A = 1/3 = .333+

Trivial.

Completely symmetric.

5.

A = 3 - 2√2 = .171+

Proved by Royce Peng, 1989.

One of an infinite family of solutions.

Horizontally symmetric.

6.
A = 1/8 = .125

Found by L. Yang, J. Z. Zhang, and Z. B. Zeng, 1991.

One of an infinite family of solutions.

Horizontally symmetric.

7.
A = 7/72 = .0972+

Found by David Cantrell, July 2006.

120o Rotationally symmetric.

8.


A = .0677+

Found by David Cantrell, July 2006.

Not symmetric.

9.

A = 43/784 = .0548+

Found by David Cantrell, July 2006.

120o Rotationally symmetric.

10.


A = .0433+

Found by David Cantrell, June 2007.

Not symmetric.

11.

A = .0360+

Found by David Cantrell, July 2006.

Horizontally symmetric.

12.
A = .0310+

Found by David Cantrell, July 2006.

Completely symmetric.

13.



A = .0265+

Found by Peter Karpov, December 2015.

Not symmetric.

14.



A = .0237+

Found by David Cantrell, June 2007.

Not symmetric.

15.

A = .0210+

Found by David Cantrell, June 2007.

120o Rotationally symmetric.

16.

A = .0179+

Found by David Cantrell, June 2007.

120o Rotationally symmetric.
More information is available at Mathworld.