Pythagorean theorem/proof

The proof of Pythagorean theorem was found by a Greek mathematician, Eudoxus of Cnidus.

The proof uses three lemmas:

1. Triangles with the same base and height have the same area.
2. A triangle which has the same base and height as a side of a square has the same area as a half of the square.
3. Triangles with two congruent sides and one congruent angle are congruent and have the same area.

The proof is:

1. Blue triangle has the same area as the green triangle, because it has the same base and height (lemma 1).
2. Green and red triangles both have two sides equal to sides of the same squares, and an angle equal to a straight angle (an angle of 90 degrees) plus an angle of a triangle, so they are congruent and have the same area (lemma 3).
3. Red and yellow triangles' areas are equal because they have the same heights and bases (lemma 1).
4. Blue triangle's area equals area of yellow triangle's area, because

${\color{blue}A_{blue}}={\color{green}A_{green}}={\color{red}A_{red}}={\color{yellow}A_{yellow}}$

1. Brown triangles have the same area for the same reasons.
2. Blue and brown each have a half of the area of a smaller square, sum of their areas equals half of the area of the bigger square, so halves of the areas of small squares are the same as a half of the area of the bigger square, so their area is the same as the area of the bigger square.