lean_workbook_plus_47221

we have $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}-6=\frac{a^2b+ab^2+a^2c+ac^2+b^2c+bc^2-6abc}{abc}$ . The nominator is nonnegative because we have $\frac{a^2b+ab^2+a^2c+ac^2+b^2c+bc^2}{6}\geq\sqrt[6]{(abc)^6}=abc$ by the AM-GM inequality.