A ninth-power rational sum at fixed total three

Let $a, b, c \in ]0, +\infty[$, such that $a + b + c = 3$, prove that:

$\frac{a^9}{(b+a)(a+c)}+\frac{b^9}{(c+b)(b+a)}+\frac{c^9}{(b+c)(c+a)} \geq \frac{3}{4}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_21765 (Apache-2.0). Natural-language proposition preserved; the erroneous original Lean transcription is corrected: Restore the product of two pair sums in each denominator; the original Lean transcription multiplies by the second factor. Proof developed independently.