A proof "by rearrangement" of the Pythagorean theorem
The Pythagorean theorem, or Pythagoras' theorem is a relation among the three sides of a right triangle (rightangled triangle).
In terms of areas, it states:
In any right triangle, the area of the square whose side is the "hypotenuse" (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two "legs" (the two sides that meet at a right angle).


There exist hundreds of proofs of this theorem. Some of these proofs use formulas that at the end lead to the Pythagorean equation.
Other proofs compare areas without using formulas. The most simple of these proofs use a rearrangement of triangles, squares,...
Perhaps, the following one is one of the simplest proofs: it's sufficient to "look":

The large square on the left consists of the square on the hypothenuse and four of the original rectangular triangles, the same large square on the right is composed of the squares on the two other sides of the rectangular triangle and four of the original rectangular triangles. The animation shows how the rearrangement of the triangles is done.
The area of the red square on the left equals the sum of the areas of two red squares on the right. And that's what Pythagoras' theorem states!
I met the idea for the animation on this wikipedia page, where
it was put in it's original form by John Blackburne
Herman Serras, July 2011
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