Cross-Figure Vertex Pinning: The Remote Triangle Anchor Effect

This technical note develops a precise formulation of the Remote Triangle Anchor Effect, a shared-vertex constraint phenomenon in Euclidean plane geometry. The central claim is modest: a triangle attached to a concurrent ray cluster does not contribute two new local facts at the shared vertex; rather, its internally determined apex angle contributes one imported angle constraint at that vertex. In the reference configuration, a right angle and a 50° angle determine a 40° apex angle, and that fixed value can then enter the neighboring ray cluster as a rank-one constraint. Under non-degeneracy and independence assumptions, this reduces by exactly one the number of additional local angle labels needed for unique determination relative to a ray-only count. The note formulates the effect in linear-algebra terms, proves the rank-one reduction statement, gives a worked triangle example, relates the idea to prior work in full-angle theorem proving and sparse linear angle methods, and presents a practical solver heuristic: before declaring a ray cluster underspecified, check whether an attached subfigure sharing the same vertex already determines an angle there. The contribution is best understood as a careful explanatory framing and bookkeeping principle for cross-figure constraint import, not as a claim of fundamentally new Euclidean mathematics.