A Study of Expansion Mandart Inellipse of A Triangle

This study was based on the research results conducted as an R&E project for gifted students with financial support from the Korea Foundation for the Advancement of Science and Creativity. This study investigates the Expansion Mandart Inellipse of a triangle, defined in terms of the concepts of the inscribed circle and the circumscribed circle, as opposed to the Mandart Inellipse defined in terms of the inscribed ellipse and circumscribed ellipse of the triangle. Through this study, the following results were obtained: Firstly, the existence and uniqueness of the Expansion Mandart Inellipse of a triangle were established. Secondly, a condition was discovered wherein the area of the inscribed circle of the triangle and the corresponding Expansion Mandart Inellipse are equal. Thirdly, regarding the intersection of the inscribed circle of the triangle and the Expansion Mandart Inellipse corresponding to it, it was found that a triangle with the three intersection points as vertices is similar to the medial triangle of the original triangle, with one of the intersection points being the center of similarity. Research like this, which extend mathematical concepts into more generalized domains, are expected to contribute to the advancement of mathematics.