Coupled linear conditions

Solve the system
$$ax+by+cz=0,a^{2}x+b^{2}y+c^{2}z=0,$$
$$x+y+z+(b-c)(c-a)(a-b)=0,$$
for $x,y,z$ in terms of $a,b,c$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. I. Ratio, Examples I, No. 26