A squared cyclic ratio sum with a symmetric quadratic correction

Proved

WorkbookSource.plus_30382

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

lean-workbooksource-checked

prove that for $a, b, c > 0$ , the following inequality holds: $\left (\frac{a}{b+c}\right)^{2}+\left (\frac{b}{c+a}\right)^{2}+\left (\frac{c}{a+b}\right)^{2}+\frac{ab+bc+ca}{a^{2}+b^{2}+c^{2}}\geq \frac{7}{4}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_30382 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_30382 (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 + (a * b + b * c + c * a) / (a ^ 2 + b ^ 2 + c ^ 2) ≥ 7 / 4   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_30382; Apache-2.0

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