DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-27

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.

Geodesic segments, geodesic triangles, and geodesic metric spaces

Definition

Let $(X, d)$ be a metric space.

A geodesic segment from $x$ to $y$ is an isometric map

$γ ⁣ : [0, d (x, y)] \to X$

with $γ (0) = x$ and $γ (d (x, y)) = y$ .

A geodesic triangle in $X$ is the union of three chosen geodesic segments joining three points $x, y, z \in X$ pairwise.

The metric space $X$ is a geodesic metric space if every pair of points is joined by at least one geodesic segment.

Depends on

Geodesics and geodesic metric spaces

Used by

Delta-slim triangles and hyperbolic spaces Definition

Dependency tree · two levels

2 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

Clara Löh, Geometric Group Theory, Section 6.2.1 (standard reference, not scraped)