A quadratic equality excludes a nondegenerate triangle

Proved

WorkbookSource.plus_81434

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Given $a^2+b^2+c^2=2(ab+bc+ca)$ , show that $a$ , $b$ , and $c$ do not form the sides of a triangle.

Source: InternLM Lean-Workbook, record lean_workbook_plus_81434 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_81434 (a b c : ℝ) (hx: a^2 + b^2 + c^2 = 2 * (a * b + b * c + c * a)) : ¬(a > 0 ∧ b > 0 ∧ c > 0 ∧ a + b > c ∧ a + c > b ∧ b + c > a)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_81434; Apache-2.0

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