Pythagorean Theorem, Hardly Knew Em

It’s a simple tattoo, but oh so important in the world of math. And it’s on a super geek. (He even sent a latexed version of the proof). For those that aren’t familiar, it’s a pictorial proof of the Pythagorean Theorem / Means Inequality.

Here’s Sam’s explanation:

Let a, b and c be the sides of the right triangle. The area of the middle square is clearly c^2. However, that is also the large square ((a+b)^2), removing the four triangles (ab/2 each). Subtracting, you get a^2+b^2.

For the inequality, the big square ((a+b)^2) is larger than the four rectangles (ab each), as the tiny square has area. Taking the square root and dividing gives (a+b)/2 > sqrt{ab}. (Note that when a=b, the square in the middle goes to 0, and we have equality.)