A shifted squared quadratic ratio sum is at least three halves

Proved

WorkbookSource.base_47449

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

lean-workbooksource-checked

Let $a, b, c>0$ . Prove that $\frac{(a^2-bc+a)^2}{b^2+c^2}+\frac{(b^2-ca+b)^2}{c^2+a^2}+\frac{(c^2-ab+c)^2}{a^2+b^2}\ge\frac{3}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_47449 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_47449 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 - b * c + a)^2 / (b^2 + c^2) + (b^2 - c * a + b)^2 / (c^2 + a^2) + (c^2 - a * b + c)^2 / (a^2 + b^2) ≥ 3 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_47449; Apache-2.0

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