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This paper presents a novel geometric approach to investigating the Riemann Hypothesis through the analysis of a specially constructed recurrence relation. We introduce a geometric framework based on triangular constructions and the cosine law, which leads to a sequence whose convergence properties are intimately connected to the distribution of zeros of the Riemann zeta function. While this work does not constitute a complete proof of the Riemann Hypothesis, it provides new geometric insights and establishes interesting connections between geometric convergence and the critical line Re(s) = 2 1 . Our analysis demonstrates how geometric methods can illuminate the deep structure underlying one of mathematics’ most famous unsolved problems.