A squared ratio sum bounds a symmetric quadratic ratio

Proved

WorkbookSource.base_17037

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

lean-workbooksource-checked

Let $a, b, c$ be positive reals. Prove that: $(\frac{a}{b+c})^2+(\frac{b}{c+a})^2+(\frac{c}{a+b})^2\ge\frac{3}{4}(\frac{a^2+b^2+c^2}{ab+bc+ca})$

Source: InternLM Lean-Workbook, record lean_workbook_17037 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_17037 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :  (a / (b + c)) ^ 2 + (b / (c + a)) ^ 2 + (c / (a + b)) ^ 2 ≥ 3 / 4 * (a ^ 2 + b ^ 2 + c ^ 2) / (a * b + b * c + a * c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_17037; Apache-2.0

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