A pairwise sum times squared reciprocals is at least nine quarters

Proved

WorkbookSource.base_855

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

lean-workbooksource-checked

Prove that for any positive real numbers $a, b, c$ ,

(ab + bc + ac) \left( \frac{1}{(a + b)^2} + \frac{1}{(b + c)^2} + \frac{1}{(a + c)^2} \right) \geq \frac{9}{4}

Source: InternLM Lean-Workbook, record lean_workbook_855 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_855 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * b + b * c + c * a) * (1 / (a + b) ^ 2 + 1 / (b + c) ^ 2 + 1 / (a + c) ^ 2) ≥ 9 / 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_855; Apache-2.0

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