This 12-page note develops a rigorous geometric theory of the Deca-Metallic Ratios and establishes their direct structural connection with primitive Pythagorean triples. Starting from the geometric construction n = 2a/(c−b), it derives clean algebraic, trigonometric, half-angle (cot(θ/2)), and Euclid-parametrization forms. The work provides a complete bidirectional characterization and classifies the associated triples into three arithmetic families (1-2-8 phenomenon). This preprint is a direct geometric companion to the algebraic theory developed in the first paper:Rajput, C. (2026). Deca-Metallic Ratios: A Base-10 Analogue of Metallic Means (Version V1). Zenodo. https://doi.org/10.5281/zenodo.19154597 It also builds upon the author’s earlier work: “Metallic Means and Pythagorean Triples” (NNTDM, 2024).
Chetansing Rajput (Thu,) studied this question.