Abstract:
An old theorem of Alexander Soifer's is the following: Given five points in a triangle of unit area, there must exist some three of them which form a triangle of area $1/4$ or less. It is easy to check that this is not true if "five" is replaced by "four", but can the theorem be improved in any other way? We discuss in this article two different extensions of the original result.
First, we allow the value of "small", $1/4$, to vary. In particular, our main result is to show that given five points in a triangle of unit area, then there must exist some three of them determining a triangle of area $6/25$ or less. Second, we put bounds on the minimum number of small triangles determined by $n$ points in a triangle, and make a conjecture about the asymptotic right answer as $n \to \infty$.
[NOTE: Conjecture 2.6 fails --- the optimal value is $3 - 2 \sqrt{2}$ not $1/6$.]
