If $$b + c = ax, c + a = by, a + b = cz$$, then the value $$\frac{1}{9}\left[\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}\right]$$ is:

$$b + c = ax, c + a = by, a + b = cz$$

x =$$\frac{b + c}{a}$$

y =$$\frac{c + a}{b}$$

z =$$\frac{a + b}{c}$$

Now,

$$\frac{1}{9}\left[\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}\right]$$

x + 1 = $$\frac{b + c}{a}$$ + 1 = $$\frac{a + b + c}{a}$$

y + 1 = $$\frac{c + a}{b}$$ + 1 = $$\frac{a + b + c}{b}$$

z + 1 = $$\frac{a + b}{c}$$ + 1 = $$\frac{a + b + c}{c}$$

= $$\frac{1}{9}\left[\frac{1}{\frac{a + b + c}{a}}+\frac{1}{\frac{a + b + c}{b}}+\frac{1}{\frac{a + b + c}{c}}\right]$$

$$\frac{1}{9}\left[\frac{a}{a + b + c}+\frac{b}{a + b + c}+\frac{c}{a + b + c}\right]$$

$$\frac{1}{9}[\frac{a + b + c}{a + b + c}]$$ = 1/9