Proving the Pythagorean Theorem

I was reading Bertrand Russell's The Problems of Philosophy on the train the other day, and that got me thinking about how universal claims, like in mathematics, can be made and justified. Though the relationship is a bit tenuous, I started thinking about how theory, the application of general principles and rules to particular instances, started with the Greeks, culminating with the genius of Pythagoras, Plato, Aristotle and Archimedes.

No matter what your level of math literacy, you know the Pythagorean theorem, the idea that the square of the hypotenuse of a right triangle equals the sum of the squares of its legs, but do you know how to prove it?

Drawing on historical analysis, late mathematician Jacob Bronowski helps you understand how the theorem can be proven, and some of its theoretical and practical significance. This is how mathematics should be taught in schools...

There are numerous ways of proving this theorem. The following is as water-tight a proof as they get :)

You will be seeing a lot more of Bronowski in the weeks to come.

In the meantime, your assignment is to figure out different ways to prove the Pythagorean Theorem. Let me know what you can come up with, and no cheating!
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