\(\frac{\left(ax+by+cz\right)^2}{x^2+y^2+z^2}=a^2+b^2+c^2\)
\(\Leftrightarrow\left(ax+by+cz\right)^2=\left(a^2+b^2+c^2\right)\left(x^2+y^2+z^2\right)\)
\(\Leftrightarrow\left(ax+by+cz\right)^2=a^2x^2+a^2y^2+a^2z^2+b^2x^2+b^2y^2+b^2z^2+c^2x^2+c^2y^2+c^2z^2\)
Mà \(\left(ax+by+cz\right)^2=a^2x^2+b^2y^2+c^2x^2+2abxy+2acxz+2bcyz\)
Nên \(a^2y^2+a^2z^2+b^2x^2+b^2z^2+c^2x^2+c^2y^2=2abxy+2acxz+2bcyz\)
\(\Leftrightarrow a^2y^2+a^2z^2+b^2x^2+b^2z^2+c^2x^2+c^2y^2-2abxy-2acxz-2bcyz=0\)
\(\Leftrightarrow\left(a^2y^2-2abxy+b^2x^2\right)+\left(a^2z^2-2acxz+c^2x^2\right)+\left(b^2z^2-2bcyz+c^2y^2\right)=0\)
\(\Leftrightarrow\left(ay-bx\right)^2+\left(az-cx\right)^2+\left(bz-cy\right)^2=0\)
\(\Rightarrow\hept{\begin{cases}ay-bx=0\\az-cx=0\\bz-cy=0\end{cases}\Rightarrow\hept{\begin{cases}ay=bx\\az=cx\\bz=cy\end{cases}\Rightarrow}\hept{\begin{cases}\frac{a}{x}=\frac{b}{y}\\\frac{a}{x}=\frac{c}{z}\\\frac{b}{y}=\frac{c}{z}\end{cases}}}\)
\(\Rightarrow\frac{a}{x}=\frac{b}{y}=\frac{c}{z}\) (đpcm)
