$\begin{array}{l} \dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 0\\ \Leftrightarrow \dfrac{{xbc + yac + zab}}{{abc}} = 0\\ \Leftrightarrow bcx + acy + zab = 0\\ + \dfrac{a}{{bc{x^2}}} + \dfrac{b}{{ac{y^2}}} + \dfrac{c}{{ab{z^2}}}\\ = \dfrac{{{a^2}{y^2}{z^2} + {b^2}{x^2}{z^2} + {c^2}{x^2}{y^2}}}{{abc{x^2}{y^2}{z^2}}}\\ = \dfrac{{\dfrac{{{a^2}}}{{{x^2}}} + \dfrac{{{b^2}}}{{{y^2}}} + \dfrac{{{c^2}}}{{{z^2}}}}}{{abc}} = \dfrac{{{{\left( {\dfrac{a}{x} + \dfrac{b}{y} + \dfrac{c}{z}} \right)}^2} - 2\left( {\dfrac{{ab}}{{xy}} + \dfrac{{bc}}{{yz}} + \dfrac{{ca}}{{zx}}} \right)}}{{abc}}\\ = \dfrac{{4 - 2.\dfrac{{abz + bcx + bac}}{{xyz}}}}{{abc}} = \dfrac{4}{{abc}} \end{array}$
