三角形中的点迫使小三角形
Points in a triangle forcing small triangles
摘要：
Alexander Soifer的一个老定理是这样的：给定一个单位面积三角形中的五个点，一定存在其中的三个点构成一个面积小于等于1/4的三角形。如果用“四”代替“五”，很容易证明这是不正确的，但是这个定理还能用其他方法改进吗？在这篇文章中，我们讨论了两个不同的扩展的原始结果。首先，我们允许“小”的值1/4变化。特别地，我们的主要结果是证明了给定单位面积三角形中的五个点，则一定存在其中的三个点决定面积小于等于6/25的三角形.其次，我们给出了由三角形中的n个点所决定的小三角形的最小数目的界，并对n趋于无穷大时的渐近正确答案作了一个猜想.
英文摘要：
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.
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