Short answers, pulled from the story.
The Pythagorean theorem states that in any right triangle, the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides. Written as an equation, the square of the hypotenuse c equals the sum of the squares of legs a and b.
Pythagoras, born around 570 BC, is credited by tradition but scholars cannot confirm he discovered the theorem. The Mesopotamian tablet Plimpton 322 and the Egyptian Berlin Papyrus 6619, both written around 1800 BC, show knowledge of the underlying relationships more than a thousand years before Pythagoras was born. Thomas L. Heath noted that no surviving Greek text from the five centuries after Pythagoras explicitly names him as the discoverer.
The book The Pythagorean Proposition catalogues 370 separate proofs, and the theorem may have more known proofs than any other in mathematics. Proofs range from geometric rearrangements and dissections to algebraic arguments, similarity of triangles, area-preserving shear mappings, and differential calculus.
Hippasus of Metapontum, dated to around 470 BC, is associated with the discovery that the Pythagorean theorem produces irrational lengths, specifically the square root of two as the hypotenuse of an isosceles right triangle with legs of length one. According to legend he was drowned at sea for making this finding public, as it contradicted the Pythagorean school's belief that all numbers are whole.
The distance formula in Cartesian coordinates is derived directly from the Pythagorean theorem. In any number of dimensions, the Euclidean distance between two points equals the square root of the sum of squared differences in each coordinate. The squared version of this distance, which avoids the square root, forms the basis of least squares methods in statistics and optimization.
No. In spherical geometry, a right triangle whose two legs each have length pi divided by two also has a hypotenuse of that same length, which violates the Pythagorean theorem. The theorem is equivalent to Euclid's fifth postulate, meaning any geometry that rejects that postulate also rejects the theorem. In very small triangles on a sphere or in hyperbolic space, the relationship approaches the Pythagorean form as the triangle shrinks toward a point.