A pair-product ratio sum bounds a normalized squared total

Let $a,b,c>0$ . Prove that:

$\frac{{2{{\left( {a + b + c} \right)}^2}}}{{{a^2} + {b^2} + {c^2}}} \le \frac{{\left( {a + b} \right)\left( {a + c} \right)}}{{{a^2} + bc}} + \frac{{\left( {b + c} \right)\left( {b + a} \right)}}{{{b^2} + ca}} + \frac{{\left( {c + a} \right)\left( {c + b} \right)}}{{{c^2} + ab}}$

Source: InternLM Lean-Workbook, record lean_workbook_21613 (Apache-2.0). Complete source proposition preserved; proof developed independently.