A Study on Siebeck-Marden’s Theorem of Convex Quadrilaterals

Abstract

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. In this study, we extended Siebeck-Marden’s Theorem, which holds for triangles, into arbitrary convex quadrilaterals. Through this study, the following results were obtained. First, we found and proved the ratio in which the inellipse of a parallelogram divides each side. Second, we discovered Siebeck-Marden’s Theorem for exellipses. We discovered Siebeck-Marden’s Theorem regarding the location of the foci of an exellipse of a triangle. Third, we defined and proved Siebeck-Marden’s Theorem of arbitrary convex quadrilaterals. We defined Siebeck-Marden’s Theorem regarding the location of the foci of an inellipse of an arbitrary convex quadrilateral. In this study, we extended Siebeck-Marden’s Theorem from triangles to convex quadrilaterals. It is expected to contribute to the development of mathematics by enhancing mathematical concepts and properties, just as we extended Siebeck-Marden’s Theorem to arbitrary convex quadrilaterals. Furthermore, this study is expected to encourage further research on the inellipse of convex quadrilaterals.