Pythagorean theorem

Statement

Let $ABC$ be a triangle in the Euclidean space $\mathbb{R}^n$. Let $d$ be the Euclidean distance, and let $\angle ABC$ be the angle at $B$. Then $$d(A, C)^2 = d(A, B)^2 + d(B, C)^2$$ if and only if $\angle ABC = \pi / 2$.

Remarks

The direction "if" is the classical theorem: in a right triangle, the square of the hypotenuse is the sum of the squares of the legs, $a^2 + b^2 = c^2$. The direction "only if" is its converse. In vector form: for vectors $u$ and $v$ with $u \cdot v = 0$, $\|u + v\|^2 = \|u\|^2 + \|v\|^2$. The theorem is the special case $\gamma = \pi / 2$ of the law of cosines. Mathlib states the equivalence for all points $A$, $B$ and $C$, with its convention for the angle at a repeated point.

History

The theorem carries the name of the Greek philosopher Pythagoras. Euclid's Elements proves it as Proposition 47 of Book I, and the converse as Proposition 48.

Explanations

L1 human

Build a square on each side of a right triangle. The two small squares together have the same area as the big square on the longest side. For the sides $3$, $4$ and $5$: $9 + 16 = 25$.