有關畢氏定理的資料

The Pythagorean theorem is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery, although knowledge of the theorem almost certainly predates him. The theorem is known in China as the "Gougu theorem" (勾股定理) for the (3, 4, 5) triangle.

The theorem is as follows:

In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (i.e. the two sides other than the hypotenuse).

If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation

圖片參考：http://upload.wikimedia.org/math/3/a/e/3ae71ab3eb71d3d182a3b9e437fba6ee.png
or, solved for c:

圖片參考：http://upload.wikimedia.org/math/4/a/a/4aa6666beea833a914b1710d8b9bcc1c.png
.

This equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of cosines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them. If the angle between the sides is right it reduces to the Pythagorean theorem.History
The history of the theorem can be divided into three parts: knowledge of Pythagorean triples, knowledge of the relationship between the sides of a right triangle, and proofs of the theorem.

Megalithic monuments from 4000 BC in Egypt, and in the British Isles from circa 2500 BC, incorporate right triangles with integer sides. Bartel Leendert van der Waerden conjectures that these Pythagorean triples were discovered algebraically.

Written between 2000–1786 BC, the Middle Kingdom Egyptian papyrus Berlin 6619 includes a problem whose solution is a Pythagorean triple.

During the reign of Hammurabi, the Mesopotamian tablet Plimpton 322, written between 1790 and 1750 BC, contains many entries closely related to Pythagorean triples.

The Baudhayana Sulba Sutra, the dates of which are given variously as between the 8th century BCE and the 2nd century BCE, in India, contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometrical proof of the Pythagorean theorem for an isosceles right triangle.

The Apastamba Sulba Sutra (circa 600 BC) contains a numerical proof of the general Pythagorean theorem, using an area computation. Van der Waerden believes that "it was certainly based on earlier traditions". According to Albert B?rk, this is the original proof of the theorem, and Pythagoras copied it on his visit to India. Many scholars find van der Waerden and B?rk's claims unsubstantiated.

Pythagoras, whose dates are commonly given as 569–475 BC, used algebraic methods to construct Pythagorean triples, according to Proklos's commentary on Euclid. Proklos, however, wrote between 410 and 485 AD. According to Sir Thomas L. Heath, there is no attribution of the theorem to Pythagoras for five centuries after Pythagoras lived. However, when authors such as Plutarch and Cicero attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted.

Around 400 BC, according to Proklos, Plato gave a method for finding Pythagorean triples that combined algebra and geometry. Circa 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented.

detail

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery,[1] although knowledge of the theorem almost certainly predates him. The theorem is known in China as the "Gougu theorem" (勾股定理) for the (3, 4, 5) triangle.


圖片參考：http://upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/180px-Pythagorean.svg.png



圖片參考：http://en.wikipedia.org/skins-1.5/common/images/magnify-clip.png
The 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).
The theorem is as follows:

In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (i.e. the two sides other than the hypotenuse).

If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation


圖片參考：http://upload.wikimedia.org/math/3/a/e/3ae71ab3eb71d3d182a3b9e437fba6ee.png

or, solved for c:


圖片參考：http://upload.wikimedia.org/math/4/a/a/4aa6666beea833a914b1710d8b9bcc1c.png

This equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of cosines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them.
This theorem may have more known proofs than any other. The Pythagorean Proposition, a book published in 1940, contains 367 proofs of Pythagoras' theorem.
The converse of the theorem is also true:

For any three positive numbers a, b, and c such that a² + b² = c², there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b.

We can also use this theorem to determine whether a triangle is right, obtuse, or acute, as follows.
If
圖片參考：http://upload.wikimedia.org/math/3/a/e/3ae71ab3eb71d3d182a3b9e437fba6ee.png
, then the triangle is right.
If c^2 \," src="http://upl oad.wikimedia.org/ma th/4/3/e/43e0c51d7e5 e078d0fc9d98c91eac09 2.png">, then the triangle is acute.
If
圖片參考：http://upload.wikimedia.org/math/a/7/0/a7022eba05a7df232b9cc2bd64405de7.png
, then the triangle is obtuse.