This triangle is super typical. There are a couple of interesting things about it, but most Geometry classes would ask kids to prove that the four small triangles are congruent.

But that smaller triangle is special, because it’s formed by connecting the midpoints of the larger triangle. It looks just like the big triangle.

Take a look at a similar situation, but with the 7-gon:

You get another 7-gon by connecting the consecutive midpoints, and it’s regular as well. (Why does it have to be regular?)

How much smaller is the inner 7-gon than the larger 7-gon? How does that compare to the difference in size with the 3-gon? What about a 4-gon?

(I got a formula for the n-gon, I think. It involves some square roots, some 1/2’s and cos(360/n).)

I like the math here — I like generalizing. But the larger message for me is that we lose out a lot by neatly dividing up Geometry into the study of triangles, quadrilaterals, circles, etc. The math that I was playing with above is about polygons, and the tired, overstretched triangle gets re-energized by it’s place as just one of many n-gons.

(By the way, in the limit, what happens to the ratio between a polygon and it’s “child”?)