(True story: I tried to read through this and found it very hard going. Then I found this playlist and was able to understand it all.)
Euclid lays out some definitions and postulates and then works and works and works with them. He proves various relationships between angles and lines and eventually provides a proof for the Pythagorean theorem. If you follow along, you'll understand each of the steps.
I'd always assumed that the Pythagorean theorem was basically found in nature and then worked out later. I thought that someone found the 3/4/5 resonance and then worked backwards from there. Now I don't know that. This proof suggests that it could have gone the other way where the it was worked out in theory first and then proved in practice. That's kind of mind-blowing to me.
As the proof started, I wanted to take out a ruler and measure each square and then do the math, but in the theoretical world that Euclid worked in, that isn't necessary. Working from already proved processes and postulates, he shows what must be true of every right triangle. What must be true even if you don't have a ruler handy.
It's hard to overstate the influence that Euclid had on the math world. From what I understand (and I'm way out of my comfort zone here), it wasn't until the 19th century that non-Euclidean math was developed. That means for more than 2000 years, math in the Western world was all based on rules and assumptions that Euclid laid down.
That's simply amazing.