\(\dfrac{\left(ax+by+cz\right)^2}{x^2+y^2+x^2}=a^2+b^2+c^2\)
\(\Leftrightarrow\left(x^2+y^2+x^2\right)\left(a^2+b^2+c^2\right)=\left(ax+by+cz\right)^2\)\(\Leftrightarrow a^2x^2+b^2x^2+c^2x^2+b^2x^2+b^2y^2+b^2z^2+c^2x^2+c^2y^2+x^2z^2=a^2x^2+b^2y^2+c^2z^2+2axby+2axcz+2bycz\)\(\Leftrightarrow\left(a^2y^2+2axby+b^2x^2\right)+\left(a^2z^2+2axcz+c^2x^2\right)+\left(b^2z^2+2bycz+c^2y^2\right)=0\)\(\Leftrightarrow\left(ay+bx\right)^2+\left(az+cx\right)^2+\left(bz+cy\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}ay=bx\\az=cx\\bz=cy\end{matrix}\right.\Leftrightarrow}\left\{{}\begin{matrix}\dfrac{a}{x}=\dfrac{b}{y}\\\dfrac{a}{x}=\dfrac{c}{z}\\\dfrac{b}{y}=\dfrac{c}{z}\end{matrix}\right.\Leftrightarrow\dfrac{a}{x}=\dfrac{b}{y}=\dfrac{c}{z}\left(đpcm\right)\)