Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Formula
a^2 + b^2 = c^2
Definition
The Pythagorean theorem says that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse: $a^2 + b^2 = c^2$, so $c = \sqrt{a^2 + b^2}$ and a missing leg is $a = \sqrt{c^2 - b^2}$, valid only for right triangles. It is equivalent to the Euclidean parallel postulate and holds only in Euclidean geometry; on a sphere, $a^2 + b^2 \neq c^2$, replaced by the spherical law of cosines. In an inner product space, $||a + b||^2 = ||a||^2 + ||b||^2$ holds if and only if $a$ and $b$ are orthogonal, a generalization to infinite dimensions.
Example
A right triangle has legs $3$ and $4$: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$, so the hypotenuse is $5$. A $6$-$8$-? right triangle gives $c = \sqrt{36 + 64} = 10$. The theorem is the foundation of the distance formula, $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$, and generalizes in $\mathbb{R}^n$: if vectors $v_1, \ldots, v_k$ are mutually orthogonal, $||v_1 + \ldots + v_k||^2 = ||v_1||^2 + \ldots + ||v_k||^2$, fundamental to Parseval's theorem in Fourier analysis.
Key Insight
Imagine squares drawn on each side of the triangle: the area of the big square on the hypotenuse exactly equals the combined area of the two smaller squares. Over $370$ proofs of the theorem are known, including one published by President James Garfield in 1876 using trapezoids, and its deep algebraic consequence is that $\mathbb{Z}[i]$ (the Gaussian integers) forms a Euclidean domain, with norms $a^2 + b^2$.