A quadratic-form bound on a sphere-plane intersection

Proved

WorkbookSource.plus_76896

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

lean-workbooksource-checked

Let $a,b,c$ be real numbers such that $a^{2}+b^{2}+c^{2}=1$ and $2a+2b-3c=1$ . Prove that:

20a^{2}+25b^{2}+13c^{2}-12ab+16bc+24ca \ge 25

Source: InternLM Lean-Workbook, record lean_workbook_plus_76896 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_76896 (a b c : ℝ) (ha : a^2 + b^2 + c^2 = 1) (hb : 2 * a + 2 * b - 3 * c = 1) : 20 * a^2 + 25 * b^2 + 13 * c^2 - 12 * a * b + 16 * b * c + 24 * c * a ≥ 25   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_76896; Apache-2.0

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