A Study on the Exellipse of Convex n-gons

In this study, we extend the concept of exellipses, which are defined in triangles and convex quadrilaterals, to convex n-gons, and define the exellipse of a convex n-gon. Through this study, the following results were obtained. First, we defined the exellipse of a convex n-gon. Based on the definitions and properties of exellipses in triangles and convex quadrilaterals, we proposed a definition of the exellipse for convex n-gons and showed that this definition is consistent with those in the cases of triangles and convex quadrilaterals. Second, we proved that for every natural number n≥3, there exists a unique exellipse corresponding to each side of a convex n-gon, and that these exellipses are similar to the inellipse of the convex n-gon. Third, we established the ratios of the major axes and the areas of the exellipses of a convex n-gon. Fourth, we determined the area of the inellipse and the exellipses of a convex n-gon. Fifth, we investigated the relationships among the lengths of the major axes and among the areas of the exellipses. It is expected that such mathematical extensions of existing concepts to broader contexts will contribute to the advancement of mathematics.