Abstract
This study addresses the inherent limitations of traditional Euclidean approaches to geometric extremum and distance problems, which frequently result in unsystematic and difficult-to-generalize solutions. To overcome these challenges, a structured three-step mathematical modeling framework is introduced: (1) vectorization of the geometric configuration; (2) formulation of an analytical objective function; and (3) systematic optimization via derivatives or gradients. Methodologically, the research employs a constructive modeling strategy that translates complex spatial relationships into vector-based algebraic expressions, rigorously tested across eight purposively selected examples ranging from elementary plane geometry to advanced spatial dynamics. The results indicate that this framework effectively transforms intricate geometric constraints into tractable, variable-dependent functions, allowing derivative-based optimization to yield precise extremum values while bypassing ad hoc geometric constructions. Ultimately, this research contributes to the literature by bridging synthetic geometry and mathematical analysis, offering a robust, generalizable tool that clarifies the underlying mathematical structures and fosters the development of abstract thinking and generalization capabilities for future pedagogical applications.