x^2-yz/a=y^2-zx/b=z^2-xy/c

Lời giải:

Đặt \(\frac{{{x^2} - yz}}{a} = \frac{{{y^2} - zx}}{b} = \frac{{{z^2} - xy}}{c} = k\,\,\,\,\left( {k \ne 0} \right)\)

\(\begin{array}{l}suy\,\,ra\left\{ \begin{array}{l}a = \frac{{{x^2} - yz}}{k}\\b = \frac{{{y^2} - xz}}{k}\\c = \frac{{{z^2} - xy}}{k}\end{array} \right.\\n\^e n\,\,\frac{{{a^2} - bc}}{x} = \frac{{{x^3} + {y^3} + {z^3} - 3xyz}}{{{k^2}}}\end{array}\)

Chứng minh tương tự:

\(\begin{array}{l}\frac{{{b^2} - ac}}{y} = \frac{{{x^3} + {y^3} + {z^3} - 3xyz}}{{{k^2}}}\\\frac{{{c^2} - bc}}{z} = \frac{{{x^3} + {y^3} + {z^3} - 3xyz}}{{{k^2}}}\\suy\,\,ra\,\,\frac{{{a^2} - bc}}{x} = \frac{{{b^2} - ca}}{y} = \frac{{{c^2} - ab}}{z}\,\,\,\left( {dpcm} \right)\end{array}\)