proof of generalization of the parallelogram law

Let $g(x,y)={\parallel x+y\parallel }^2-{\parallel x\parallel }^2$ and $m(x,y)=⟨x,y⟩+⟨y,x⟩$.Then

$$g(x,y)={\parallel y\parallel }^2+m(x,y).$$

Hence, taking $x_1=x_4=x,x_2=y,x_3=z$ we have:

${\displaystyle \sum _{i=1}^3}{\parallel x_i+x_{i+1}\parallel }^2-{\displaystyle \sum _{i=1}^3}{\parallel x_i\parallel }^2$$=$${\displaystyle \sum _{i=1}^3}g(x_i,x_{i+1})$

$=$${\displaystyle \sum _{i=1}^3}{\parallel x_i\parallel }^2+{\displaystyle \sum _{i=1}^3}m(x_i,x_{i+1})$

$=$${\parallel {\displaystyle \sum _{i=1}^3}{x}_i\parallel }^2.$