Từ: $\dfrac{a}{x}+\dfrac{b}{y}+\dfrac{c}{z}=0$
$\Leftrightarrow \dfrac{ayz+bxz+cxy}{xyz}=0$
$\Leftrightarrow ayz+bxz+cxy=0$
Từ: $\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1$
$\Leftrightarrow \dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}+\dfrac{{{z}^{2}}}{{{c}^{2}}}+\dfrac{2xy}{ab}+\dfrac{2yz}{bc}+\dfrac{2xz}{ac}=1$
$\Leftrightarrow \dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}+\dfrac{{{z}^{2}}}{{{c}^{2}}}+\dfrac{2cxy}{abc}+\dfrac{2ayz}{abc}+\dfrac{2bxz}{abc}=1$
$\Leftrightarrow \dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}+\dfrac{{{z}^{2}}}{{{c}^{2}}}+\dfrac{2}{abc}\left( ayz+bxz+cxy \right)=1$
$\Leftrightarrow \dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}+\dfrac{{{z}^{2}}}{{{c}^{2}}}=1$
