The Pythagorean theorem is one of the fundamental theorems of Euclidean geometry , establishing the relationship between the sides of a right triangle : the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse .
The ratio in one form or another was supposedly known to various ancient civilizations long before our era; the first geometric proof is attributed to Pythagoras . The statement appears as Proposition 47 in Euclid 's Elements.
It can also be expressed as a geometric fact that the area of the square built on the hypotenuse is equal to the sum of the areas of the squares built on the legs. The converse statement is also true : a triangle, the sum of the squares of the lengths of two sides of which is equal to the square of the length of the third side, is a right triangle.
There are a number of generalizationsohis theorem - for arbitrary triangles , for figures in spaces of higher dimensions. In non-Euclidean geometries, the theorem does not hold.
The main formulation contains algebraic operations - in a right-angled triangle, the lengths of the legs of which are equal a and b, and the length of the hypotenuse is c, the relation:
a2+b2=c2
