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)