Let $u=x\operatorname{cis}(A), v=y\operatorname{cis}(B), w=z\operatorname{cis}(C)$, and let $a_n=\sum u^n$. Clearly $a_1, a_2\in\mathbb{R}$, and it follows that \[\sum uv=\frac{1}{2}(a_1^2-a_2)\in\mathbb{R}.\] We have $uvw=xyz\cdot e^{i(A+B+C)}=\pm xyz,$ since $A+B+C=k\pi$ for some $k\in\mathbb{Z}$. Thus we can construct some cubic $p^3+ap^2+bp+c\in\mathbb{R}[p]$ with roots $u, v, w$. If we let $p=u,v,w$ and add the results we get \[a_3+a\cdot a_2+b\cdot a_1+3c=0,\] and it follows that $a_3\in\mathbb{R}$. Now, multiplying by $p^n$ and proceeding as above yields \[a_{n+3}+a\cdot a_{n+2}+b\cdot a_{n+1}+c\cdot a_n=0,\] and it follows by induction that $a_n\in\mathbb{R}$ for all positive $n$, and the result follows.