DefinitionDefinition: AI-adaptedProof: 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.

Base and perpendicular height for a chosen side of a plane figure

Definition

Let $p, v, w \in R^{2}$ with $v \neq 0$ . For the parallelogram $P (p; v, w)$ (Parallelograms and triangles in $R^{2}$ ), the base length along $v$ is $∥ v ∥_{2}$ , and the corresponding perpendicular height is

$d (w, R v) : = \inf_{t \in R} ∥ w - t v ∥_{2}, R v : = {t v : t \in R} .$

The infimum is the Euclidean point-to-set distance; it is defined because the line $R v$ is nonempty (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

For the triangle $T (A, B, C)$ with chosen nonzero side $A B$ , the base length is $∥ B - A ∥_{2}$ and the corresponding height is $d (C - A, R (B - A))$ . This is the distance from $C$ to the full line through $A$ and $B$ , not necessarily to the segment $[A, B]$ .

Depends on

Used by

Dependency tree · two levels

28 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)