A symmetric quadratic ratio bounds a pairwise cyclic sum

Proved

WorkbookSource.plus_20950

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

lean-workbooksource-checked

For positive real numbers $a,b,c$ prove that $\frac{a^2+b^2+c^2}{ab+bc+ca}\ge \frac 23\left(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\right)$

Source: InternLM Lean-Workbook, record lean_workbook_plus_20950 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_20950 (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) ≥ 2 / 3 * (a / (a + b) + b / (b + c) + c / (c + a))   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_20950; Apache-2.0

View graph