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In mathematics, the Pythagorean theorem or Pythagoras's theorem, is a relation in Euclidean geometry between the three sides of a right triangle. The theorem is named after and commonly attributed to the 6th century BC Greek philosopher and mathematician Pythagoras, although the facts of the theorem were known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived.

1 The theorem

The Pythagorean theorem states:

The sum of the areas of the squares on the legs of a right triangle is equal to the area of the square on the hypotenuse.

A right triangle is a triangle with one right angle; the legs are the two sides that make up the right angle, and the hypotenuse is the third side opposite the right angle. In the picture below, a and b are the legs of a right triangle, and c is the hypotenuse:

Pythagoras perceived the theorem in this geometric fashion, as a statement about areas of squares:

The sum of the areas of the blue and red squares is equal to the area of the purple square.

Using algebraAlgebra Elementary algebra is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. While in arithmetic only numbers and their arithmetical operations (such as +,, one can reformulate the theorem into its modern expression by noting that the area of a square is the square (second power) of the length of its side:

Given a right triangle with legs of lengths a and b and hypotenuse of length c, then a² + b² = c².

2 A visual proof

Perhaps this theorem has a greater variety of different known proofs than any other (the law of quadratic reciprocityIn mathematics, the law of quadratic reciprocity in number theory, conjectured by Euler and Legendre and first satisfactorily proved by Gauss, connects the solvability of two related quadratic equations in modular arithmetic. As a consequence, it allows u may also be a contender for that distinction).

This illustration depicts one of them. In the right half of the picture, four copies of this triangle surround a large square. The pink diagonal square in the center is the square on the hypotenuse. Move the four triangles within the large square so that they are arranged as in the left half of the picture. Then the pink area not included within the four triangles makes up the squares on the legs. Consequently the sum of the areas of the squares on the legs equals the area of the square on the hypotenuse. Q.E.D.For other meanings of the abbreviation "QED", see QED. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, "that which was to be demonstrated"). This is a translation of the Greek oper edei deixai which was used by many early mathem

NB: This proof is very simple, but it is not elementary, in the sense that it does not depend solely upon the most basic axioms and theorems of Euclidean geometry. In particular, while it is easy to give a formula for area of triangles and squares, it is not as easy to prove that the area of a square is the sum of areas of its pieces. In fact, proving the necessary properties is harder than proving the Pythagorean theorem itself. For this reason, axiomatic introductions to geometry usually employ another proof based on the similarity of triangles (see proof 6 in the external link).

There are many different proofs of the Pythagorean theorem; one was developed by United States President James Garfield. One of the proofs is based on Euler's formula in complex analysis. (See also the external links below for a sampling of the many different proofs of the Pythagorean theorem.)

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Topics: Pythagorean Theorem Triangle Geometry Euclidean Square Form...

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