A cyclic quadratic ratio sum bounds the normalized quadratic mean

Proved

WorkbookSource.base_8926

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers. Prove that:

\frac{a^2}{b^2+2ab}+\frac{b^2}{c^2+2bc}+\frac{c^2}{a^2+2ca} \geq \frac{3(a^2+b^2+c^2)}{(a+b+c)^2}

C-S with $(a+c)^2$ helps, but a full expanding gives something obvious.

Source: InternLM Lean-Workbook, record lean_workbook_8926 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_8926 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 / (b^2 + 2 * a * b) + b^2 / (c^2 + 2 * b * c) + c^2 / (a^2 + 2 * c * a)) ≥ 3 * (a^2 + b^2 + c^2) / (a + b + c)^2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_8926; Apache-2.0

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