Given: \[ a^2 = by + cz \Rightarrow a^2 + ax = ax + by + cz \Rightarrow a(a + x) = ax + by + cz \Rightarrow a + x = \frac{ax + by + cz}{a} \] Similarly, \[ b^2 = cz + ax \Rightarrow b^2 + by = ax + by + cz \Rightarrow b(b + y) = ax + by + cz \Rightarrow b + y = \frac{ax + by + cz}{b} \] And, \[ c^2 = ax + by \Rightarrow c^2 + cz = ax + by + cz \Rightarrow c(c + z) = ax + by + cz \Rightarrow c + z = \frac{ax + by + cz}{c} \] Now, \[ \frac{x}{a + x} + \frac{y}{b + y} + \frac{z}{c + z} = \frac{x}{\frac{ax + by + cz}{a}} + \frac{y}{\frac{ax + by + cz}{b}} + \frac{z}{\frac{ax + by + cz}{c}} = \frac{ax}{ax + by + cz} + \frac{by}{ax + by + cz} + \frac{cz}{ax + by + cz} \] \[ = \frac{ax + by + cz}{ax + by + cz} = 1 \quad \text{(Proved)} \]