A Study on Napoleon’s Theorem for Convex n-gons

In this study, Napoleon’s theorem, originally defined in triangles, is extended to convex polygons with n sides, where n is a natural number greater than or equal to 4. Through this research, the following results were obtained. First, it was revealed that the Napoleon point of a convex n-gon is related to the Fermat-Torricelli point within the convex n-gon. Second, the properties of the Napoleon n-gon derived from a convex n-gon were clarified. It was shown that the Napoleon n-gon is always a similar n-gon, and its side lengths, area, and perimeter were determined. Third, the properties of the regular Napoleon n-gon were established. The necessary and sufficient conditions for a Napoleon n-gon to be regular were identified, and a formula for its area was derived. It is expected that such activities, which expand mathematical concepts into broader domains, will contribute to the advancement of mathematics.