# The Heilbronn Problem for Circles

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

# 3.

A = 3√3 / 4 = 1.299+

Trivial.

Completely symmetric.

# 4.

A = 1

Trivial.

Completely symmetric.

# 5.

A = √(50-10√5) / 8 = .657+

Trivial.

Completely symmetric.

# 6.

A = √3 / 4 = .433

Trivial.

Completely symmetric.

# 7.

A = .294+

Trivial.

Completely symmetric.

# 8.

A = .216+

Found by David Cantrell, July 2006.

Completely symmetric.

# 9.

A = .173+

Found by David Cantrell, July 2006.

Horizontally symmetric.

# 10.

A = .150+

Found by David Cantrell, July 2006.

Completely symmetric.

# 11.

A = .113+

Found by David Cantrell, August 2006.

Horizontally symmetric.

# 12.

A = .104+

Found by David Cantrell, July 2006.

Symmetry of an equilateral triangle.

# 13.

A = .0856+

Found by David Cantrell, June 2007.

Horizontally symmetric.

# 14.

A = .0758+

Found by David Cantrell, June 2007.

Horizontally symmetric.

# 15.

A = .0700+

Found by David Cantrell, August 2006.

Horizontally symmetric.

# 16.

A = .0661+

Found by David Cantrell, August 2006.

Symmetry of a square.