Triangle geometry/Pythagorean theorem/Fact/Proof

We denote the vertices of the triangle by ${\displaystyle {}A,B,C}$, with the right angle located at ${\displaystyle {}C}$. We set ${\displaystyle {}v=C-A}$, and ${\displaystyle {}w=B-C}$. The vector from ${\displaystyle {}A}$ to ${\displaystyle {}B}$ equals ${\displaystyle {}v+w}$, and ${\displaystyle {}v}$ and ${\displaystyle {}w}$ are perpendicular to each other. Therefore,

${\displaystyle {}\Vert {v+w}\Vert ^{2}=\left\langle v+w,v+w\right\rangle =\left\langle v,v\right\rangle +2\left\langle v,w\right\rangle +\left\langle w,w\right\rangle =\Vert {v}\Vert ^{2}+\Vert {w}\Vert ^{2}\,.}$