In any right-angled triangle, the square on the hypotenuse (the side opposite the right angle) equals the sum of the squares on the other two sides. If the legs are a and b, and the hypotenuse is c, then a² + b² = c². A 3-4-5 triangle satisfies 9 + 16 = 25.
a² + b² = c². For the 3-4-5 triangle: 9 + 16 = 25. The blue and red squares together equal the green square in area.
Babylonian clay tablets from 1900 BC list Pythagorean triples (3,4,5), (5,12,13), (8,15,17), showing the result was known empirically long before Pythagoras. His school (around 570 BC) gave the first proof. Over 370 different proofs are now known, including algebraic, geometric, trigonometric, and one published by US President James Garfield in 1876.
In n dimensions: the distance from the origin to (x₁, x₂, …, xₙ) is √(x₁² + x₂² + ⋯ + xₙ²). Fermat's Last Theorem (proved by Andrew Wiles in 1995 after 358 years) shows there are no integer solutions to aⁿ + bⁿ = cⁿ for n greater than 2. The Pythagorean theorem is the n=2 case with infinitely many integer solutions.
Both big squares are (a+b)×(a+b). Both contain four identical right triangles. What is left over in the left square is c². What is left over in the right square is a²+b². They must be equal.
In any right triangle: a^2 + b^2 = c^2. Known empirically to Babylonians by 1800 BC; first proved by the Pythagoreans around 570 BC. Over 370 distinct proofs exist, including one by US President James Garfield in 1876. Integer solutions are Pythagorean triples: all triples are generated by (m^2-n^2, 2mn, m^2+n^2). Fermat's Last Theorem (proved by Wiles, 1995) shows no analogous integer solutions exist for exponents above 2. The theorem extends to n dimensions as the Euclidean distance formula.