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More than 4000 years ago, the Babyloneans and the Chinese already knew that a triangle with the sides of 3, 4 and 5 must be a right triangle. They used this knowledge to construct right angles. By dividing a string into twelve equal pieces and then laying it into a triangle so that one side is three, the second side four and the last side five sections long, they could easily construct a right angle.
A Greek scholar named Pythagoras, who lived around 500 BC, was also fascinated by triangles with these special side ratios. He studied them a bit closer and found that the two shorter sides of the triangles squared and then added together, equal exactly the square of the longest side. And he proved that this doesn't only work for the special triangles, but for any right triangle. Today we would write it somehow like this: a2 + b2= c2. In the time of Pythagoras they didn't use letters yet to replace variables. (They weren't introduced until the 16th century by Vieta.) Instead they wrote down everything in words, like this: if you have a right triangle, the squares of the two sides adjacent to the right angle will always be equal to the square of the longest side.
We can't be sure if Pythagoras really was the first person to have found this relationship between the sides of right triangles, since no texts written by him were found. In fact, we can't even prove the guy lived. But the theorem a2 + b2= c2 got his name. Another Greek, Euclid, wrote about the theorem about 200 years later in his book called "Elements". There we also find the first known proof for the theorem. Now there are about 600 different proofs.
Today the Pythagorean theorem plays an important part in many fields of mathematics. For example, it is the basis of Trigonometry, and in its arithmetic form it connects Geometry and Algebra.
The "Leafed" and the "Confierous" Pythagorean Trees are a nice connection between the Pythagorean theorem and fractal geometry. You can read more about them in the Mandelbrot and Sierpinski page.
