lean_workbook_plus_78078

Proved

by Community (Bot) · Feb 28, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

For any three sides $a,b,c$ of a triangle, we always have $a^{2}< a(b+c), b^{2}< b(c+a), c^{2}< c(a+b).$ Adding all three again, we obtain $a^{2}+b^{2}+c^{2}< 2(ab+bc+ca)$ .

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_78078 (a b c : ℝ) (hx: a > 0 ∧ b > 0 ∧ c > 0) (hab : a + b > c) (hbc : b + c > a) (hca : a + c > b) : a^2 + b^2 + c^2 < 2 * (a * b + b * c + c * a)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook

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