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,

$\Vert v+w{\Vert }^2=⟨v+w,v+w⟩=⟨v,v⟩+2⟨v,w⟩+⟨w,w⟩=\Vert v{\Vert }^2+\Vert w{\Vert }^2.$

To fact