A Study on the Steiner Inellipse and Marden’s Theorem of Quadrilaterals

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. A Steiner inellipse, defined in a triangle, is the maximum-area inellipse of the triangle, and the ratio of the area of the triangle and its Steiner inellipse is constant. Also, the Steiner inellipse satisfies Marden’s theorem. In this study, we expanded the Steiner inellipse, which was defined in triangles, to a quadrilateral and researched its existence and properties—along with the absence of Marden’s Theorem. Through this study, the following results were obtained. First, we found that a quadrilateral in which its Steiner inellipse exists is a parallelogram. Second, we discovered that the Steiner inellipse of a quadrilateral is the maximum-area inellipse of the quadrilateral. Thus, we proved that there was a constant ratio between the area of the Steiner inellipse and the area of the quadrilateral. Third, we showed that Marden’s theorem of quadrilaterals holds. That is, the relationship between the four vertices of the quadrilateral and the two focal points of the Steiner inellipse was found. Fourth, we unveiled a method of drawing the Steiner inellipse of a given quadrilateral. It is expected to contribute to the development of mathematics by expanding mathematical concepts just as we expanded the Steiner inellipse—which was only defined in triangles—to quadrilaterals. In addition, it is expected that further research on the expansion of the Steiner inellipse will be actively carried out through this study.