Pythagorean theorem

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.