If \[ \frac{by + cz}{b^2 + c^2} = \frac{cz + ax}{c^2 + a^2} = \frac{ax + by}{a^2 + b^2} \] \[ = \frac{by + cz + cz + ax + ax + by}{b^2 + c^2 + c^2 + a^2 + a^2 + b^2} \] [By addition process] \[ = \frac{2(ax + by + cz)}{2(a^2 + b^2 + c^2)} = \frac{ax + by + cz}{a^2 + b^2 + c^2} \] \[ \therefore \frac{by + cz}{b^2 + c^2} = \frac{ax + by + cz}{a^2 + b^2 + c^2} \] \[ \Rightarrow (by + cz)(a^2 + b^2 + c^2) = (b^2 + c^2)(ax + by + cz) \] \[ \Rightarrow a^2(by + cz) + (b^2 + c^2)(by + cz) = ax(b^2 + c^2) + (b^2 + c^2)(by + cz) \] \[ \Rightarrow a^2(by + cz) = ax(b^2 + c^2) \] \[ \Rightarrow \frac{by + cz}{b^2 + c^2} = \frac{ax}{a^2} \] \[ \Rightarrow \frac{by + cz}{b^2 + c^2} = \frac{x}{a} \quad \text{—— (i)} \] \[ \frac{cz + ax}{c^2 + a^2} = \frac{ax + by + cz}{a^2 + b^2 + c^2} \] \[ \Rightarrow (cz + ax)(a^2 + b^2 + c^2) = (c^2 + a^2)(ax + by + cz) \] \[ \Rightarrow b^2(cz + ax) + (c^2 + a^2)(cz + ax) = by(c^2 + a^2) + (c^2 + a^2)(cz + ax) \] \[ \Rightarrow b^2(cz + ax) = by(c^2 + a^2) \] \[ \Rightarrow \frac{cz + ax}{c^2 + a^2} = \frac{by}{b^2} \] \[ \Rightarrow \frac{cz + ax}{c^2 + a^2} = \frac{y}{b} \quad \text{—— (ii)} \] \[ \frac{ax + by}{a^2 + b^2} = \frac{ax + by + cz}{a^2 + b^2 + c^2} \] \[ \Rightarrow (ax + by)(a^2 + b^2 + c^2) = (a^2 + b^2)(ax + by + cz) \] \[ \Rightarrow c^2(ax + by) + (a^2 + b^2)(ax + by) = cz(a^2 + b^2) + (a^2 + b^2)(ax + by) \] \[ \Rightarrow c^2(ax + by) = cz(a^2 + b^2) \] \[ \Rightarrow \frac{ax + by}{a^2 + b^2} = \frac{cz}{c^2} \] \[ \Rightarrow \frac{ax + by}{a^2 + b^2} = \frac{z}{c} \quad \text{—— (iii)} \] From (i), (ii), and (iii), \[ \frac{x}{a} = \frac{y}{b} = \frac{z}{c} \quad \text{(Proved)} \]