A symmetric quadratic ratio bounds cyclic difference ratios

Proved

WorkbookSource.plus_40002

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

lean-workbooksource-checked

Given $a, b, c > 0$ . Prove that: $\frac {a^2 + b^2 + c^2}{ab + bc + ca} \geq\ \frac {a^2 + bc - ca}{a^2 + ab + bc} + \frac {b^2 + ca - ba}{b^2 + bc + ca} + \frac {c^2 + ab - bc}{c^2 + ca + ab}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_40002 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_40002 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b^2 + c^2) / (a * b + b * c + c * a) ≥ (a^2 + b * c - c * a) / (a^2 + a * b + b * c) + (b^2 + c * a - a * b) / (b^2 + b * c + c * a) + (c^2 + a * b - b * c) / (c^2 + c * a + a * b)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_40002; Apache-2.0

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