This paper presents an independent geometric derivation involving linked right triangles and algebraic substitution based on the Pythagorean theorem. The derivation develops a relation for determining an unknown side length by connecting multiple right-triangle configurations through shared variables. By applying the Pythagorean theorem to related geometric structures and performing algebraic elimination, an explicit expression for an unknown side is obtained. The work emphasizes geometric visualization, structural reasoning, and step-by-step analytical derivation rather than the introduction of a new theorem. A numerical example is provided to verify the validity of the derived relation and its consistency with the original geometric configuration. This exposition demonstrates how classical geometric principles may be reconstructed through independent mathematical reasoning and provides an educational illustration of algebraic-geometric problem solving within linked right-triangle systems.