A Study on the Extension of Napoleon’s Theorem for Convex Hexagons

This study was based on the research results conducted as a R&E project for the gifted students with a financial support from the Korea Foundation for the Advancement of Science and Creativity. In this study, the Napoleon hexagon ofhexagons was explored by expanding the Fermat-Torricelli point defined in a triangle and the Napoleon triangle into a convex hexagon. The results of this study are as follows. First, the equivalence condition for convex hexagons in which the Napoleon hexagon exists was discovered. Second, the area formula for Napoleon’s hexagon was shown. Third, we found various regular hexagons related to the Napoleon-containing hexagon and found that the centers of each regular hexagon were identical. Through this study, the concepts of the Fermat-Torricelli point and Napoleon triangle of triangles were expanded to convex hexagons, and it is expected that the expansion of mathematical concepts willcontribute to the development of mathematics. In addition, it is expected that additional research will be conducted in the future by exploring attractive regular hexagons whose centers coincide with the searched hexagons including Napoleon.