Points in a triangle forcing small triangles

TL;DR

This paper explores extensions of a theorem about points in a triangle, showing that five points guarantee a small triangle of area at most 6/25, and discusses bounds and conjectures for larger point sets.

Contribution

The paper extends an existing theorem by allowing the small triangle area to vary and provides bounds and conjectures for the minimum number of small triangles for larger point sets.

Findings

Five points in a triangle of unit area guarantee a triangle of area ≤ 6/25

Bounds on the minimum number of small triangles for n points are discussed

A conjecture about the asymptotic behavior as n increases

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 tends to infinity.