After 40 years of dealing with math, I know found the true reason for the Pythagoras result on this page. I admit that I did not know this.
Here is a sketch:
ACB is given as a rectangular triangle. Then it is easy to see that the green, the brown, and the triangle ABC are similar. So the area of each of these triangles is proportional to the square of its hypothenus (the long side) with the same factor of proportion. And obviously the blue and brown area add to the total area. That’s it:
\(a^2+b^2 = c^2\)
In fact, the Pythagoras theorem is proved as soon, as it is proved for any three similar figures on the sides. To prove it for squares is just a special case. We can just as well take similar triangles as we did in the proof above.
Another famous example of similar figures are half circles.
The two blue areas add to the brown area. This allows to compute the area of the following Apollonian moons. It is equal to the area of the triangle.