A quadratic-product lower bound for triangle sides

Proved

WorkbookSource.plus_77465

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

lean-workbooksource-checked

Let $a$ , $b$ and $c$ be the lengths of the sides of a triangle such that $a + b + c = 3$ . Prove that $4\left(a^2+b^2+c^2\right)+abc \geq 13.$

Source: InternLM Lean-Workbook, record lean_workbook_plus_77465 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_77465 (a b c : ℝ) (hx: a + b + c = 3) (hab : a + b > c) (hbc : b + c > a) (hca : a + c > b) : 4 * (a ^ 2 + b ^ 2 + c ^ 2) + a * b * c ≥ 13   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_77465; Apache-2.0

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