2017 Putnam B5

A line in the plane of a triangle $T$ is called an *equalizer* if it divides $T$ into two regions having equal area and equal perimeter. Find positive integers $a>b>c$, with $a$ as small as possible, such that there exists a triangle with side lengths $a, b, c$ that has exactly two distinct equalizers.

如果三角形 $T$ 平面中的一条线将 $T$ 分成面积和周长相等的两个区域，则该线称为*均衡器*。找到正整数 $a>b>c$，其中 $a$ 尽可能小，这样就存在一个边长为 $a、b、c$ 的三角形，并且恰好有两个不同的均衡器。