This study explores whether, given the area of an outer triangle, one can deduce the side length and area of the original configuration in regular 𝑛-gons and inscribed polygon constructions. The research establishes a process that interconverts geometric and algebraic representations, deriving recursive sequences through area ratios and constructing higher-order equations. New notations, T_P^Q &〖〖 U〗_n〗_q^p, are introduced to represent non-adjacent product-sums, simplifying algebraic structures. By applying mathematical induction and extremal logic, the study successfully proves that if a higher-order equation has a positive real root, its largest positive real root corresponds uniquely to a constrained polygonal configuration. Conversely, if such a configuration exists, it uniquely corresponds to the largest positive real root of the equation. This research not only offers a systematic method for geometric inference but also establishes a clear model for interpreting algebraic equations geometrically, with theoretical significance and potential for broader application.