A Research on the Inscribed Ellipse of Kite and Regular Polygon

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. Any triangle or convex quadrilateral has an infinite number of inscribed ellipses (Agarwal, Clifford, & Lachance, 2015). In the case of triangles, there is an inscribed ellipse with the focus on any point inside (Park, Park, & Cho, 2020), while the set of points that can be the focus of the inscribed ellipse of a parallelogram forms a specific trace (Park, Park, & Cho, 2021). Therefore, in the case of convex quadrilaterals other than parallelogram, there was a question about how the traces of the focus of the inscribed ellipse were drawn, and among those quadrilateral, the nature of the focus was investigated for the inscribed ellipse of the kite. As a result, in this study, it was proved that these three propositions are equivalent to each other: the point is the focus of the inscribed ellipse; four points, that each is symmetrical to the point for one of the sides, lie on the same circle; the point is a square focus. Based on this, the traces of the focal point of the inscribed ellipse of the kite were shown to be the diagonal which is the axis of symmetry, and the arc that passes through incenter and two vertices that not on the axis of symmetry. Furthermore, using the traces, it could be proved that the only inner ellipse of the regular polygon with 5 or more sides was the incircle, which is meaningful in that it showed the uniqueness of the inscribed ellipse in a different way from the existing projective geometric approach (Agarwal, et al., 2015).