Figure 1

Marion Walter is a professor of mathematics at the University of Oregon.  This project can lead you to some of the same discoveries that she made about what happens when you trisect the sides of a triangle.

Start with any triangle and trisect all three sides.  Then connect each vertex to the trisection points on the opposite side as in Figure 1.   Here are two of several questions that can be asked about this figure.

In the Figure 1, there is a small hexagon in the middle of the triangle.  Will there always be a hexagon, no matter what the original triangle looks like?

Can you relate the area of the hexagon to the area of the original  triangle?  Can you prove the relationship?  You can create a figure like Figure 2 with geometry software.

Figure 2

---