# Pythagorean theorem

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Also known as Pythagoras' theorem, the Pythagorean theorem is a theorem in geometry that gives a relationship between the sides of a right-angled triangle. The Pythagorean theorem was evidently known before Pythagoras (6th century B.C.), but the proof in general form is ascribed to him. Originally the theorem established a relationship between the areas of the squares constructed on the sides of a right-angled triangle: The square on the hypotenuse is equal to the sum of the squares on the other sides. Sometimes, the Pythagorean theorem is formulated briefly as follows: The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the catheti. The theorem converse to the Pythagorean theorem is also true: If the square of a side of a triangle is equal to the sum of the squares of the other two sides, then that triangle is right-angled.

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The Pythagorean theorem is a special case of the cosine theorem; its infinite-dimensional analogue (in Hilbert space) is the Parseval equality (i.e. the completeness theorem for orthonormal systems).

The problem of solving Pythagoras' equation \$a^2+b^2=c^2\$ in integers \$a,b,c\$ leads to the Pythagorean numbers. The problem of solving its generalization, the Diophantine equation \$a^n+b^n=c^n\$, \$n\ge3\$, is called Fermat's last (or great) theorem, cf. Fermat great theorem.

A right-angled triangle with sides having integer lengths is called a Pythagorean triangle.

How to Cite This Entry:
Pythagoras theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pythagoras_theorem&oldid=20501