Pythagorean Theorem

The Pythagorean theorem is one of the oldest results in mathematics. For any right triangle with legs of length $a$ and $b$ and hypotenuse $c$,

$a^2 + b^2 = c^2.$

It has been proved hundreds of times — by rearrangement, by similarity, by algebra, and by calculus — and generalises to inner-product spaces in higher mathematics.

Direct applications include: computing distances in the plane (the distance formula is just the theorem in disguise), checking whether a triangle is right-angled, navigating in 2D / 3D, and laying out construction.