DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Parallelograms and triangles in $R^{2}$

Definition

For $p, v, w \in R^{2}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ), the closed parallelogram with base point $p$ and spanning vectors $v, w$ is

$P (p; v, w) : = {p + s v + t w : 0 \leq s \leq 1, 0 \leq t \leq 1} .$

For $A, B, C \in R^{2}$ , the closed triangle with vertices $A, B, C$ is

$T (A, B, C) : = {A + s (B - A) + t (C - A) : s \geq 0, t \geq 0, s + t \leq 1} .$

These definitions include singular cases: a spanning vector may vanish, and vertices may be repeated or collinear. They also satisfy $P (p; v, w) = p + P (0; v, w)$ and $T (A, B, C) = A + T (0, B - A, C - A)$ in the notation of Translation of a subset of $R^{n}$ .

Depends on

Used by

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

W. F. Trench, Introduction to Real Analysis, §7.3 (standard reference, not scraped)

Parallelograms and triangles in R2R2