lean_workbook_plus_5727

In triangle $ABC$ denote $\{\begin{array}{c} \alpha = \frac {b + c}{a} \ \ \beta = \frac {c + a}{b} \ \ \gamma = \frac {a + b}{c}\end{array}$ . Prove that $\{\begin{array}{c} \alpha + \beta + \gamma + 2 = \alpha\beta\gamma \ \ 2(\alpha + \beta + \gamma )\le \alpha\beta + \beta\gamma + \gamma\alpha\end{array}$