Generalizing the Equal Incircles Theorem: Insights from Sangaku Problems

Sangaku problems are traditional Japanese geometrical puzzles, often displayed in religious temples, that have intrigued mathematicians for centuries. This study aims to generalize the Equal Incircles Theorem, extending Angela Drei’s proof to N-circles, by applying the trigonometric method alongside foundational mathematical tools, including mathematical induction, Heron’s formula, and the telescoping product. A generalized equation for N circles based on the Equal Incircles Theorem is derived through explicit mathematical formulation and characterization. The findings deepen our understanding of geometric relationships, highlight the historical significance of Sangaku problems, and offer potential advancements for future engineering applications, mathematics education, and research in mathematical history.