The Pythagorean Theorem is one of Euclidean Geometry's most beautiful theorems. It is simple, yet obscure, and is used continuously in mathematics and physics. In short, it is really cool.
Evidence of the theorem can be traced far back into Egyptian history with the help of the Rhind Papyrus(1788-1580 BC). The Rhind Papyrus itself claims to be a copy of an earlier work, possibly dating as far back as 2000 BC. The use of the 3-4-5 triangles(9+16=25) to construct perfect right angles, indeed seems to have been a very common practice, long ago. Unfortunately, little information predates the Greeks, so this will probably remain another mystery of the Egyptians. Traditionally, however, the theorem has been credited[in western culture] to Pythagoras of Samos. The legend has it that he was so excited by its proof that he sacrificed a bull for the occasion, even though Pythagoreans were against animal sacrifice. Unfortunately, there are only legends. The Pythagorean School, which gets its name from its founder, was a secret cult. They regarded their knowledge as something to be kept from all outsiders. Thus, they did not write things down until the cult began to lose prominence several generations later, leaving posterity with a void where the life of Pythagoras should have been. Consequently, classicists do not know if Pythagoras was actually responsible for the first proof.
How does one prove this enigma? The geometry books I have had experience with turned this rose into a briar. Unfortunately, nobody knows how "Pythagoras" originally proved the theorem, but here are three ways.
This first method is one of the ways the Pythagoreans would have proved the theorem. Unfortunately, it lacks glamour. In the following picture let ABC be a right triangle and BD be a segment drawn perpendicular to AC.
Since the triangles are similar, the sides must be of proportional lengths.
