Simple geometric proof of the pythagorean theorem
This geometric construction proves the Pythagorean theorem: the sum of the squares of the sides of a right triangle equals the square of the length of the hypotenuse.
The red, green, yellow, and blue triangles are all the same. In the figure on the left, the white square clearly has area equal to the square of the hypotenuse of a colored triangle, since those hypotenuses form the sides of the square.
In the figure at the right, we have two white squares. One has area equal to A^2, where A is the length of one side of the colored triangles; and the other has area B^2, where B is the length of the other side of the colored triangles.
The total area of the figure on the left is the same as the total area of the figure on the right. The combined area of colored triangles is the same in both figures. Therefore, the area of the white square on the left (C^2) equals the sum of the areas of the white squares on the right (A^2+B^2).
A^2 + B^2 = C^2
Created using Donald Knuth's Metapost program.
( from nibot-lab.livejournal.com/32463.html )