Chứng minh rằng nếu (a^2 +b^2 +c^2) (x^2 +y^2 +z^2)

* Hướng dẫn giải

$$\begin{gathered} \left(a^2+b^2+c^2\right)\left(x^2+y^2+z^2\right)={\left(ax+by+cz\right)}^2 \\ \iff \left(a^2{y}^2-2abxy+b^2{x}^2\right)+\left(a^2{z}^2-2acxz+c^2{x}^2\right)+\left(b^2{z}^2-2bcyz+c^2{y}^2\right)=0 \\ \iff {\left(ay-bx\right)}^2+{\left(az-cx\right)}^2+{\left(bz-cy\right)}^2=0 \\ \iff ay-bx=0,az-cx=0,bz-cy=0 \\ \iff \frac{a}{x}=\frac{b}{y}=\frac{c}{z} \end{gathered}$$

Với $x,y,z\ne 0$