Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between 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.

The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation:

a

2

+

b

2

=

c

2

.

{\displaystyle a^{2}+b^{2}=c^{2}.}

The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.

When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points.

The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but n-dimensional solids.

History

Forms of the Pythagorean theorem have appeared in many ancient cultures, and the date of first discovery is uncertain, as is the date of the first proof. The history of the development of the theorem involves multiple aspects, including calculations regarding specific right triangles, knowledge of Pythagorean triples, understanding the relationship among the sides of a right triangle, and proofs of the theorem within some deductive system.

Written c.