Viviani’s theorem is this really cool proof about equilateral triangles.  I’m not sure who first proved it (it’s named for Viviani, obviously). It states that if you take any point inside the triangle, and draw the shortest straight lines to the sides, the sum of the three resulting lines is equal to the height of the triangle.

I’ve been playing around with making representations and demos in Graspable Math, a research project and free teaching tool my post-doc Erik Weitnauer, Erin Ottmar, and I are making together (all beta disclaimers apply).  I thought I’d make a quick proof of Viviani’s theorem, and I was pretty pleased with the result. If you want to play with it yourself, here’s a link to the canvas–but be forewarned, as of Dec 1, 2016, there’s some glitch with our saving and loading, which breaks some of the links. You’re better off deriving the proof yourself next to the proof that loads. I demo that here.

Viviani’s theorem has some funny implications. For instance, barycentric coordinate plots–the coolest way to plot three values constrained to a constant sum–couldn’t work without them.

These plots come up all the time in my work, because we often have tasks where subjects have to choose among three items. They are also the best way to think about how you pay your money to humble bundle: you gotta pay all your money, so the sum is fixed, but the values are ‘free’ to vary.