LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Barycentric coordinates are unique

Statement

Let $v_{0}, \dots, v_{n}$ be affinely independent points in $R^{m}$ . If $x = \sum_{i = 0}^{n} λ_{i} v_{i} = \sum_{i = 0}^{n} μ_{i} v_{i}, \sum_{i = 0}^{n} λ_{i} = \sum_{i = 0}^{n} μ_{i} = 1,$ then $λ_{i} = μ_{i}$ for every $i$ .

Proof

Given: Two barycentric-coordinate expressions for the same point of the simplex spanned by $v_{0}, \dots, v_{n}$ .

1.1given

Subtract the two expressions for $x$ to obtain $\sum_{i = 0}^{n} (λ_{i} - μ_{i}) v_{i} = 0$ , and subtract the two sum conditions to obtain $\sum_{i = 0}^{n} (λ_{i} - μ_{i}) = 0$ .

2.1step 1.1∎

The previous step is an affine dependence relation among $v_{0}, \dots, v_{n}$ whose coefficients sum to $0$ . Since the vertices are affinely independent, every coefficient vanishes, so $λ_{i} - μ_{i} = 0$ for all $i$ .

Depends on

The geometric simplex spanned by affinely independent vertices

Barycentric coordinates are unique

Statement

Proof

Depends on

Dependency tree · 1 level

Sources