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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|cosine theorem]]; its infinite-dimensional analogue (in [[Hilbert space|Hilbert space]]) is the [[Parseval equality|Parseval equality]] (i.e. the completeness theorem for orthonormal systems).

The problem of solving Phytagoras' equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075940/p0759401.png" /> in integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075940/p0759402.png" /> leads to the [[Pythagorean numbers|Pythagorean numbers]]. The problem of solving its generalization, the Diophantine equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075940/p0759403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075940/p0759404.png" />, is called Fermat's last (or great) theorem, cf. [[Fermat great theorem|Fermat great theorem]].

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.H. Greub,   "Linear algebra" , Springer  (1967)</TD></TR></table>