A cyclic product ratio bounds a symmetric quadratic ratio

Proved

WorkbookSource.base_50724

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

lean-workbooksource-checked

For any positive real numbers $a,b$ and $c$ , prove that $\frac{a(a+c)}{b(b+c)}+\frac{b(b+a)}{c(c+a)}+\frac{c(c+b)}{a(a+b)}\ge\frac{3(a^{2}+b^{2}+c^{2})}{ab+bc+ca}$

Source: InternLM Lean-Workbook, record lean_workbook_50724 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_50724 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * (a + c) / (b * (b + c)) + b * (b + a) / (c * (c + a)) + c * (c + b) / (a * (a + b))) ≥ 3 * (a ^ 2 + b ^ 2 + c ^ 2) / (a * b + b * c + c * a)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_50724; Apache-2.0

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