A pairwise-product sum times reciprocal ratios bounds a quadratic sum

Proved

WorkbookSource.base_40806

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

lean-workbooksource-checked

Prove that if $a,b,c>0$ then: $(ab+bc+ca)\left( \frac{bc}{a^2}+\frac{ca}{b^2}+\frac{ab}{c^2}\right)\ge 3(a^2+b^2+c^2)$

Source: InternLM Lean-Workbook, record lean_workbook_40806 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_40806 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * b + b * c + c * a) * (b * c / a ^ 2 + c * a / b ^ 2 + a * b / c ^ 2) ≥ 3 * (a ^ 2 + b ^ 2 + c ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_40806; Apache-2.0

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