# Pythagoras

Pythagoras(580-500 BC) was the "Great Thinker" who discovered the Pythagorean Theorem in geometry. Pythagoras was born in Samos on the western coast of what is now Turkey. He was reportedly the son of a rich citizen named Mnesarchos. There he lived for many years under the rule of the tyrant Polycrates, who had a tendency to switch alliances in times of conflict, which were frequent. He met Thales, likely as a young man, who recommended he travel to Egypt. It seems certain that he gained much of his knowledge from the Egyptians, as had Thales before him. Not much else is known about Pythagoras, other than that he was a mathematician and philosopher who founded a community in southern Italy, sometime in the 6th century. His followers were extremely secretive and loyal, and held a mystical view of numbers and their relation to nature. Very similar to monks.

The theorem states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. There are 54 proofs of the Pythagorean Theorem. Here are a couple of examples. This is probably the most famous of all proofs of the Pythagorean proposition. It's the first of Euclid's two proofs (I.47). The underlying configuration became known under a variety of names, the Bride's Chair likely being the most popular.

First of all, ABF = AEC by SAS. This is because, AE = AB, AF = AC, and
BAF = BAC + CAF = CAB + BAE = CAE.

1)We start with two squares with sides a and b, respectively, placed side by side. The total area of the two squares is a2+b2.

2)The construction did not start with a triangle but now we draw two of them, both with sides a and b and hypotenuse c. Note that the segment common to the two squares has been removed. At this point we therefore have two triangles and a strange looking shape.

3)We rotate the triangles 90 degrees, each around its top vertex. The right one is rotated clockwise whereas the left triangle is rotated counterclockwise. Obviously the resulting shape is a square with the side c and area c squared.

The 20th president of the United States gave the following proof to the Pythagorean Theorem. He discovered this proof five years before he become President. He hit discovered this proof in 1876 during a mathematics discussion with some of the members of Congress. It was later published in the New England Journal of Education. The proof depends on calculataing the area of a right trapezoid two different ways. The first way is by using the area formula of a trapezoid and the second is by adding up the areas of the three right triangles that can be constructed in the trapezoid. He used the following trapezoid in developing his proof.

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