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 = .0260+

Found by David Cantrell, June 2007.

120o Horizontally 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.

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