Ta có : \(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}\)
\(\Rightarrow\frac{abz-cya}{a^2}=\frac{bcx-abz}{b^2}=\frac{cay-bcx}{c^2}=\frac{abz-cya+bcx-abz+cay-bcx}{a^2+b^2+c^2}\)
\(\Rightarrow abz-cya=0\Leftrightarrow abz=cya\Leftrightarrow bz=cy\Leftrightarrow\frac{y}{b}=\frac{z}{c}\left(1\right)\)
\(\Rightarrow bcx-abz=0\Leftrightarrow bcx=abz\Leftrightarrow cx=az\Leftrightarrow\frac{x}{a}=\frac{z}{c}\left(2\right)\)
Từ \(\left(1\right),\left(2\right)\) Ta có : \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\)
Ta có : \(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}\)
\(\Rightarrow\frac{abz-cya}{a^2}=\frac{bcx-abz}{b^2}=\frac{cay-bcx}{c^2}=\frac{abz-cya+bcx-abz+cay-bcx}{a^2+b^2+c^2}=0\)
\(\Rightarrow abz-cya=0\Leftrightarrow abz=cya\Leftrightarrow bz=cy\Leftrightarrow\frac{y}{b}=\frac{z}{c}\)(1)
và \(bcx-abz=0\Leftrightarrow bcx=abz\Leftrightarrow cx=az\Leftrightarrow\frac{x}{a}=\frac{z}{c}\)(2)
Từ (1) và (2) ta có \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\)