If $\frac{x}{y+z}=\frac{y}{z+x}=\frac{z}{x+y}=K(K\ne 0)$, then the value of $K$ is:

By applying the properties of equal ratios:
$$K=\frac{\text{Sum of Numerators}}{\text{Sum of Denominators}}=\frac{x+y+z}{(y+z)+(z+x)+(x+y)}=\frac{x+y+z}{2(x+y+z)}$$

Case 1: If $x+y+z\ne 0$, we can cancel out the term $(x+y+z)$ from the numerator and denominator:

$$K=\frac{1}{2}$$

Case 2: If $x+y+z=0$, then $y+z=-x$. Substituting this into the first ratio gives:

$$K=\frac{x}{y+z}=\frac{x}{-x}=-1$$

Thus, the value of $K$ can be either $\frac{1}{2}$ or $-1$.