Pythagorean theorem

Pythagorean theorem
The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":[1]

where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides.

Although it is often argued that knowledge of the theorem predates him,[2][3] the theorem is named after the ancient Greek mathematician Pythagoras (c. 570–495 BC) as it is he who, by tradition, is credited with its first proof, although no evidence of this exists.[4][5][6] There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework.[7][8] Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases.

The theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps and cartoons abound.

Rearrangement proof

The rearrangement proof (click to view animation)

The two large squares shown in the figure each contain four identical triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the white space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem, Q.E.D.[9]

Heath gives this proof in his commentary on Proposition I.47 in Euclid's Elements, and mentions the proposals of Bretschneider and Hankel that Pythagoras may have known this proof. Heath himself favors a different proposal for a Pythagorean proof, but acknowledges from the outset of his discussion "that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometric discovery to him."[10] Recent scholarship has cast increasing doubt on any sort of role for Pythagoras as a creator of mathematics, although debate about this continues.[11]

Other Languages
Alemannisch: Satz des Pythagoras
azərbaycanca: Pifaqor nəzəriyyəsi
Bân-lâm-gú: Pythagoras tēng-lí
беларуская: Тэарэма Піфагора
беларуская (тарашкевіца)‎: Тэарэма Пітагора
davvisámegiella: Pythagorasa cealkka
emiliàn e rumagnòl: Tioréma 'd Pitàgora
hornjoserbsce: Sada Pythagorasa
Bahasa Indonesia: Teorema Pythagoras
lietuvių: Pitagoro teorema
Lingua Franca Nova: Teorem de Pitagora
Bahasa Melayu: Teorem Pythagoras
oʻzbekcha/ўзбекча: Pifagor teoremasi
Simple English: Pythagorean theorem
slovenčina: Pytagorova veta
slovenščina: Pitagorov izrek
srpskohrvatski / српскохрватски: Pitagorina teorema
татарча/tatarça: Pifagor teoreması
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українська: Теорема Піфагора
vepsän kel’: Pifagoran teorem
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文言: 勾股定理
吴语: 勾股定理
粵語: 勾股定理
žemaitėška: Pėtaguora teuorema
中文: 勾股定理