ta có :bz-cy/a=cx-az/a=ay-bz=bxz-cxy/ax=cxy-ayz/by=ayz-bxz/cz=0/ax+by+cz=0
=>bz=cy=>z/c=y/b (1)
cx=az=>x/a=z/c (2)
ay=bz=>y/b=x/a (3)
từ (1),(2),(3) suy ra x/a=y/b=z/c hay x:y:z=a:b:c

Biết rằng: $\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}$
Vậy ta có: $\frac{\left( bz-cy \right)x}{ax}=\frac{\left( cx-az \right)y}{by}=\frac{\left( ay-bx \right)z}{cz}$
Hay \[\frac{bzx-cxy}{ax}=\frac{cxy-ayz}{by}=\frac{ayz-bxz}{cz}=\frac{bzx-cxy+cxy-ayz+ayz-bxz}{ax+by+cz}=0\]
Vậy $\left\{ \begin{align} & bz=cy \\ & cx=az \\ & ay=bx \\ \end{align} \right.$ hay $\left\{ \begin{align} & \frac{b}{y}=\frac{c}{z} \\ & \frac{a}{x}=\frac{c}{z} \\ & \frac{a}{x}=\frac{b}{y} \\ \end{align} \right.$ hay $\frac{a}{x}=\frac{b}{y}=\frac{c}{z}.$ (điều phải chứng minh)