It seems out of order to start here, but because I started my quest for calculus a week before I started writing about it, I know for a fact I'm going to want to use the Pythagorean Theorem several times in the near future. I ended up having to research this one. I kept trying to build a proof using similar triangles (it turns out I was close, but the related proof is messy). There are several proofs out there. This is the one I like the best. I made the images in Excel, so don't judge me.
Start out with 4 identical triangles with sides a, b, and c:
Rotate the second triangle 90°, the third 180°, and the fourth 270°. Shove them all together like Pangaea making a large square with a small empty square in the middle as such:
Now, it's easy to observe that the area of the large square is 
. Alternately, we can deduce that the area of the large square is the area of the 4 triangles plus the area of the small square in the middle. The small square has an area of 
. Each of the triangles individually have an area of 
.
This means that:
Which simplifies to:
Which simplifies further to:
Which of course goes to:
Giving us the end result:
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