## Points in a triangle forcing small triangles

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Kahle, Matthew

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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.

Comment: 14 pages, 17 figures; in honor of Alexander Soifer's 60th birthday

Comment: 14 pages, 17 figures; in honor of Alexander Soifer's 60th birthday

##### Keywords

Mathematics - Combinatorics, Mathematics - Metric Geometry, 05B50