A cyclic quadratic ratio upper bound with a symmetric correction

Proved

WorkbookSource.base_26563

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

lean-workbooksource-checked

Let: $a;b;c > 0$ .Prove that: $2.\frac {a^2 + b^2 + c^2}{ab + bc + ca} \ge \frac {a^2}{c^2 + ca + ab} + \frac {b^2}{a^2 + ab + bc} + \frac {c^2}{b^2 + bc + ca} + 1$

Source: InternLM Lean-Workbook, record lean_workbook_26563 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_26563 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 2 * ((a^2 + b^2 + c^2) / (a * b + b * c + c * a)) ≥ (a^2 / (c^2 + c * a + a * b)) + (b^2 / (a^2 + a * b + b * c)) + (c^2 / (b^2 + b * c + c * a)) + 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_26563; Apache-2.0

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