cho tam giác abc chứng minh cot a + cot b + cot c =a2+b2 +c2/4S

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
\frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin C}} = 2R\\
\cos A = \frac{{{b^2} + {c^2} – {a^2}}}{{2bc}}\\
S = \frac{{abc}}{{4R}}\\
\cot A + \cot B + \cot C\\
 = \frac{{\cos A}}{{\sin A}} + \frac{{\cos B}}{{\sin B}} + \frac{{\cos C}}{{\sin C}}\\
 = \frac{{\frac{{{b^2} + {c^2} – {a^2}}}{{2bc}}}}{{\frac{a}{{2R}}}} + \frac{{\frac{{{c^2} + {a^2} – {b^2}}}{{2ca}}}}{{\frac{b}{{2R}}}} + \frac{{\frac{{{a^2} + {b^2} – {c^2}}}{{2ab}}}}{{\frac{c}{{2R}}}}\\
 = \frac{R}{{abc}}.\left( {{b^2} + {c^3} – {a^2} + {c^2} + {a^2} – {b^2} + {a^2} + {b^2} – {c^2}} \right)\\
 = \frac{1}{{4S}}.\left( {{a^2} + {b^2} + {c^2}} \right)\\
 = \frac{{{a^2} + {b^2} + {c^2}}}{{4S}}
\end{array}\)