Ta có:
\(cotA=\dfrac{cosA}{sinA}=\dfrac{\dfrac{b^2+c^2-a^2}{2bc}}{\dfrac{a}{2R}}=R.\dfrac{b^2+c^2-a^2}{abc}\)
Tương tự: \(cotB=R.\dfrac{a^2+c^2-b^2}{abc}\) ; \(cotC=R.\dfrac{a^2+b^2-c^2}{abc}\)
\(\Rightarrow cotA+cotB+cotC=\dfrac{R}{abc}\left(b^2+c^2-a^2+a^2+c^2-b^2+a^2+b^2-c^2\right)\)
\(=\dfrac{R}{abc}.\left(a^2+b^2+c^2\right)\)
Mà \(S=\dfrac{abc}{4R}\Rightarrow\dfrac{R}{abc}=\dfrac{1}{4S}\)
\(\Rightarrow cotA+cotB+cotC=\dfrac{a^2+b^2+c^2}{4S}\)
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