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	The '''Pythagorean Theorem''' or '''Pythagoras' Theorem''' is named for and attributed to the ] Greek philosopher and mathematician ], though the facts of the theorem were known before he lived. It states a relationship between the lengths of the sides of a right triangle:

	:The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse.

	(A right triangle is one with a right angle; the legs are the two sides that make up the right angle; the hypotenuse is the third side opposite the right angle).

	Visually, the theorem can be described as follows:
	Given a right triangle, with sides a and b and hypotenuse c, (Figure 1)

	,
	,'\|
	,' \|
	,' \|
	'''c''' ,' \|
	,' \| '''b'''
	,' \|
	,' ___\|
	,' \| \|
	,'____________\|___\|

	'''a'''


	Figure 1


	then it follows that '''a<sup>2</sup>+b<sup>2</sup>=c<sup>2</sup>'''.


	'''Proof:''' Draw a right triangle with sides ''a'', ''b'', and ''c'' as above. Then take a copy of this triangle and place its ''a'' side in line with the ''b'' side of the first, so that their ''c'' sides form a right angle (this is possible because the angles in any triangle add up to two right angles -- think it through). Then place the ''a'' side of a third triangle in line with the ''b'' side of the second, again in such a manner that the ''c'' sides form a right angle. Finally, complete a square of side (''a+b'') by placing the ''a'' side of a fourth triangle in line with the ''b'' side of the third. On the one hand, the area of this square is '''(a+b)<sup>2</sup>''' because (''a+b'') is the length of its sides. On the other hand, the square is made up of four equal triangles each having area ''ab/2'' plus one square in the middle of side length ''c''. So the total area of the square can also be written as '''4 · ab/2 + c<sup>2</sup>'''. We may set those two expressions equal to each other and simplify:

	:'''(a+b)<sup>2</sup> = 4 · ab/2 + c<sup>2</sup>'''

	:'''a<sup>2</sup> + 2ab + b<sup>2</sup> = 2ab + c<sup>2</sup>'''

	:'''a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup>'''

	]

	The sheer volume of distinct known proofs of this theorem is staggering. See http://www.cut-the-knot.com/pythagoras/index.html for just a sampling.

	The converse of the Pythagorean Theorem is also true:
	if you have three positive numbers '''a''', '''b''', and '''c''' such that '''a<sup>2</sup>+b<sup>2</sup>=c<sup>2</sup>''', then there exists a triangle with sides '''a''', '''b''' and '''c''', and every such triangle has a right angle between the sides '''a''' and '''b'''.
	This can be proven using the ] which is a generalization of the Pythagorean theorem applying to ''all'' triangles, not just right-angled ones.

	Another interpretation of the Pythagorean Theorem was already given by ] in his ''Elements'':
	if one erects similar figures (see ]) on the sides of a right triangle,
	then the sum of the areas of the two smaller ones equals the area of the larger one.

	Yet another generalization of the Pythagorean Theorem is ''Parseval's identity'' in ].

	One should note that the Pythagorean Theorem is a theorem in ] and follows from the axioms of that theory.
	A priori, it need not be true for actually existing triangles that we measure out in our universe.
	One of the first mathematicians to realize this was ], who actually carefully measured out huge right triangles as part of his geographical surveys in order to check the theorem.
	He found the theorem to be correct. The theory of ] holds that in huge gravitational fields, such as close to ]s, the theorem is violated.
	On much larger, cosmological scales, the theorem may very well be violated as well, as a result of a large-scale curvature of space. This is an open problem in ].


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	'''See also:'''
⚫	* ]

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	]

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Pythagorean Theorem