A pairwise ratio sum with a symmetric quadratic correction

Proved

WorkbookSource.base_39284

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

lean-workbooksource-checked

The following is also true :

$\frac {a + b}{c} + \frac {b + c}{a} + \frac {c + a}{b} + \frac {ab + bc + ca}{a^2 + b^2 + c^2}\ge 7$

with $a,b,c > 0$

Source: InternLM Lean-Workbook, record lean_workbook_39284 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_39284 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b) / c + (b + c) / a + (c + a) / b + (a * b + b * c + c * a) / (a ^ 2 + b ^ 2 + c ^ 2) ≥ 7  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_39284; Apache-2.0

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