Trapped in a Triangle

So this was a not-all-that interesting problem to me. I'll admit to having looked up the "answer" after getting nowhere with it for a while—I started down the brute-force path that was laid out in the Answers section but turned around too early, fully convinced that I was on the wrong path.

Anyhoo, to make up for the small hit to my self-esteem, I made this animation (code) hinting at the theorem referred to below. The problem reads as follows:

So Many Equations

I'll skip some steps for your sanity: \[\small\mathtt{(BD)^2 + (PD)^2 = (FB)^2 + (PF)^2}\] \[\small\mathtt{(DC)^2 + (PD)^2 = (CE)^2 + (PE)^2}\] \[\small\mathtt{(EA)^2 + (PE)^2 = (AF)^2 + (PF)^2}\]

I think you can fill in the missing links above. Then the idea is to subtract (2) from (1) and then subtract (3) from the result.

The conclusion that pops out in the end—the general theorem you are supposed to derive—is that, having divided the perimeter of the triangle into 6 segments with the perpendiculars, the sum of the squares of every other side is equal to the sum of the squares of the remaining sides.

Looking for a Real Math Headache? Learn Programming.

The most interesting part of putting any of these notebook entries together is the programming part. Right now, programming is without a doubt the best sandbox one could possibly have for learning mathematics. You've got application and exploration and, importantly, you sacrifice little in the way of abstraction in a programming environment.

This, for example, is one of the equations I needed to do the animation above:

\(\small\mathtt{d = sin(\frac{60\pi}{180} - arcsin(\frac{230 - y}{\sqrt{(x + 100)^2 + (230 - y)^2}})) \cdot \sqrt{(x + 100)^2 + (230 - y)^2}}\)

It's completely sloppy and inelegant, but it's what I could work out on paper as a first step to locating where the perpendiculars intersected the non-horizontal lines of the triangle. I needed those locations in order to know where to draw the colorful lines on the perimeter.

The downside, of course, is that everyone needs to know a little programming first, before we can use it as an environment in which to play out mathematics learning. And that's hard.

Philosophies cost money. Better to have a cheap one.