Triangle geometry/Pythagorean theorem/Fact/Proof

Proof

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

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

To fact