Pythagorasztételt

Pythagorasztételt, also known as the Pythagorean theorem, is a fundamental principle in Euclidean geometry, named after the ancient Greek mathematician Pythagoras. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed algebraically as:

a^2 + b^2 = c^2

where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides. The theorem is a direct consequence of the definition of similarity between two right triangles: two right triangles are similar if and only if their corresponding angles are congruent, and the corresponding sides are proportional. The converse of the Pythagorean theorem is also true: if the lengths of the sides of a triangle satisfy the Pythagorean theorem, then the triangle is a right triangle.

The Pythagorean theorem has numerous applications in various fields, including architecture, engineering, and physics. It is also a cornerstone of trigonometry and is used to solve problems involving right triangles. The theorem has been generalized to higher dimensions and to non-Euclidean geometries, and it has inspired numerous mathematical proofs and generalizations throughout history.