A Study on the Exellipse of Triangle

This study was based on the research results conducted as a YSC project. Studies on the inellipse of a triangle and previous studies that explored the properties between the incircle and the excircle of a triangle were analyzed. I wondered if the excircle of a triangle could be extended to an exellipse, and whether the properties that exist between an incircle and a excircle of a triangle also hold true between an inellipse and an exellipse of a triangle. Therefore, in this study, I defined the exellipse of a triangle and explored the properties. Through this study, the following research results were obtained. First, the exellipse of the triangle was defined, and its existence and uniqueness were proved. Second, we found the division ratio at which the exellipse internally and externally divides the line segment and extension line of a triangle. Third, it was revealed that various properties including the Lurier theorem for ellipses were established in triangles. Fourth, a method of constructing an exellipse of a triangle was discovered. Based on this study, it is expected that follow-up studies on the exellipse of the triangle and the expansion of the various triangle centers will be actively conducted through this study.