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).