If $\dfrac{x}{y+z} = \dfrac{y}{z+x} = \dfrac{z}{x+y}$ and are not mutually equal, the value of each ratio is—

If $x+y+z\neq0$, adding $1$ to each ratio shows every ratio equals $\frac{1}{2}$; if $x+y+z=0$, each numerator equals its negative denominator, making every ratio $-1$. Since $x,y,z$ are not all equal, both possibilities $-1$ and $\frac{1}{2}$ arise.