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+

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

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.

120^o Rotationally symmetric.

8.

A = .0677+

Found by David Cantrell, July 2006.

Not symmetric.

9.

A = 43/784 = .0548+

Found by David Cantrell, July 2006.

120^o 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 = .0260+

Found by David Cantrell, June 2007.

120^o Horizontally symmetric.

14.

A = .0237+

Found by David Cantrell, June 2007.

Not symmetric.

15.

A = .0210+

Found by David Cantrell, June 2007.

120^o Rotationally symmetric.

16.

A = .0179+

Found by David Cantrell, June 2007.

120^o Rotationally symmetric.

More information is available at Mathworld.

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