# 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.