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The proof uses three lemmas:
- Triangles with the same base and height have the same area.
- 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.
- Triangles with two congruent sides and one congruent angle are congruent and have the same area.
The proof is:
- Blue triangle has the same area as the green triangle, because it has the same base and height (lemma 1).
- 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).
- Red and yellow triangles' areas are equal because they have the same heights and bases (lemma 1).
- Blue triangle's area equals area of yellow triangle's area, because
- Brown triangles have the same area for the same reasons.
- 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.