Checking positivity of a quadratic form

Identity.
$$Q=\frac12\left[(x+y)^2+(y+z)^2+(z+x)^2\right].$$
Verify: RHS $=\frac12(2x^2+2y^2+2z^2+2xy+2yz+2zx)=x^2+y^2+z^2+xy+yz+zx=Q$.

Vanishing. $Q=0\iff x+y=0$, $y+z=0$, $z+x=0$. Adding: $2(x+y+z)=0$, so $x+y+z=0$, then combining with $x+y=0$ gives $z=0$, similarly $x=y=0$.

Conclusion. $Q\ge0$ with equality iff $(x,y,z)=(0,0,0)$, so $Q$ is positive definite.