`[bz-cy]/a=[cx-az]/b=[ay-bx]/c`
`=> [a(bz-cy)]/[a^2]=[b(cx-az)]/[b^2]=[c(ay-bx)]/[c^2]`
`=> [baz-cay]/[a^2]=[bcx-baz]/[b^2]=[cay-bcx]/[c^2]=[baz-cay+bcx-baz+cay-bcx]/[a^2+b^2+c^2]=0/[a^2+b^2+c^2]=0`
$⇒\begin{cases}\dfrac{bz-cy}{a}=0\\\dfrac{cx-az}{b}=0\end{cases}$
$⇒\begin{cases}bz-cy=0\\cx-az=0\end{cases}$
$⇒\begin{cases}bz=cy\\cx=az\end{cases}$
$⇒\begin{cases}\dfrac{z}{c}=\dfrac{y}{b}\\\dfrac{x}{a}=\dfrac{z}{c}\end{cases}$
`=> x/a = y/b = z/c`
`=> x : y : z = a : b : c`
