Áp dụng tính chất của dãy tỉ số bằng nhau ta có:
\(\dfrac{bz-cy}{a}=\dfrac{cx-az}{b}=\dfrac{ay-bx}{c}\)
\(=\dfrac{a\left(bz-cy\right)}{a^2}=\dfrac{b\left(cx-az\right)}{b^2}=\dfrac{c\left(ay-bx\right)}{c^2}\)
\(=\dfrac{abz-acy}{a^2}=\dfrac{bcx-abz}{b^2}=\dfrac{acy-bcx}{c^2}\)
\(=\dfrac{abz-acy+bcx-abz+acy-bcx}{a^2+b^2+c^2}\)
\(=\dfrac{0}{a^2+b^2+c^2}=0\)
\(\Rightarrow abz-acy=bcx-abz=acy-bcx\)
\(\Rightarrow a\left(bz-cy\right)=b\left(cx-az\right)=c\left(ay-bx\right)\)
\(\Rightarrow bz-cy=cx-az=ay-bx\)
\(\Rightarrow\left\{{}\begin{matrix}bx=cy\\cx=az\\ay=bx\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}\dfrac{x}{c}=\dfrac{y}{b}\\\dfrac{x}{a}=\dfrac{z}{c}\\\dfrac{y}{b}=\dfrac{x}{a}\end{matrix}\right.\Rightarrow\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}\)
Vậy \(x:y:z=a:b:c\)
