A symmetric quadratic reciprocal lower bound

Proved

WorkbookSource.base_39998

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

lean-workbooksource-checked

Given $x,y,z>0$ ,prove that: ${\frac { \left( x+y+z \right) ^{2}}{{x}^{2}+2\,yz}}+{\frac { \left( x+y+z \right) ^{2}}{{y}^{2}+2\,xz}}+{\frac { \left( x+y+z \right) ^{2}}{ {z}^{2}+2\,xy}}\geq 9$

Source: InternLM Lean-Workbook, record lean_workbook_39998 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_39998 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x + y + z) ^ 2 / (x ^ 2 + 2 * y * z) + (x + y + z) ^ 2 / (y ^ 2 + 2 * x * z) + (x + y + z) ^ 2 / (z ^ 2 + 2 * x * y) ≥ 9  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_39998; Apache-2.0

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