A quadratic ratio sum has a symmetric lower bound

Proved

WorkbookSource.plus_67114

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers. Prove the inequality $\frac{a^2}{b^2+bc+c^2}+\frac{b^2}{c^2+ca+a^2}+\frac{c^2}{a^2+ab+b^2}\ge\frac{1}{9}\cdot\frac{a^2+b^2+c^2}{ab+bc+ca}+\frac{8}{9}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_67114 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_67114 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 / (b^2 + b * c + c^2) + b^2 / (c^2 + c * a + a^2) + c^2 / (a^2 + a * b + b^2)) ≥ 1 / 9 * (a^2 + b^2 + c^2) / (a * b + b * c + a * c) + 8 / 9   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_67114; Apache-2.0

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