Cho $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ và $\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 0$. Chứng minh rằng: $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$.

Ta có:

+) $\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 0 \Leftrightarrow \frac{{ayz + bxz + cxy}}{{xyz}} = 0 \Leftrightarrow ayz + bxz + cxy = 0$

+) $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \Leftrightarrow {\left( {\frac{x}{a} + \frac{y}{b} + \frac{z}{c}} \right)^2} = 1$

$\Leftrightarrow \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} + 2\left( {\frac{{xy}}{{ab}} + \frac{{yz}}{{bc}} + \frac{{xz}}{{zc}}} \right) = 1$

$\Leftrightarrow \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} + 2\left( {\frac{{ayz + bxz + cxy}}{{abc}}} \right) = 1$

$\Leftrightarrow \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$ (đpcm).