2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67974Let F be a subfield of a commutative field extending R. Let ϕ_2: F^2 \times F^2 \to F, ϕ_2((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:R^2 \to F^2 preserves distance d \geq 0 if for each x,y \in R^2 |x-y|=d implies ϕ_2(f(x),f(y))=d^2. We prove that each unit-distance preserving mapping f:R^2 \to F^2 has a form I \circ (ρ,ρ), where ρ: R \to F is a field homomorphism and I: F^2 \to F^2 is an affine mapping with orthogonal linear part.LaTeX2e, 10 pagesMetric Geometry51M05text