You have
$\left\{ \begin{align*} &(a+b+c)(b-a)=1\\ &(a+b+c)(c-a)=2\\ &(a+b+c)(c-b)=1\;. \end{align*}\right.\tag{1}$
These clearly imply that $a+b+c\ne 0$, so the first and third of these imply that $b-a=c-b$. In other words, $\langle a,b,c\rangle$ is an arithmetic progression. (The second equation of $(1)$ confirms this.) Set $d=b-a$; then $b=a+d$ and $c=a+2d$, so $a+b+c=3(a+d)$, and each of the equation in $(1)$ reduces to $3d(a+d)=1$.
Going back to the original equations, we see that
$\begin{align*} 3&=a^2-bc=a^2-(a+d)(a+2d)\\ &=-3ad-2d^2=-3d(a+d)+d^2\\ &=d^2-1\;, \end{align*}$
or $d^2=4$. Thus, $d=\pm 2$, $1=3d(a+d)=\pm6(a+d)$, and $a+b+c=3(a+d)=\pm\frac12$.