A symmetric quadratic ratio with a pair-product correction

Proved

WorkbookSource.base_26631

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

lean-workbooksource-checked

If $a,b,c>0$ then prove that

$\frac{a^2+b^2+c^2}{ab+bc+ca}+\frac{8(ab+bc+ca)(a+b+c)}{9(a+b)(b+c)(c+a)}\geq 2$

Source: InternLM Lean-Workbook, record lean_workbook_26631 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_26631 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b^2 + c^2) / (a * b + b * c + a * c) + (8 * (a * b + b * c + a * c) * (a + b + c)) / (9 * (a + b) * (b + c) * (c + a)) ≥ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_26631; Apache-2.0

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