A Study on the Lemoine Inellipse of Convex Quadrilaterals

This study extends the concept of the Lemoine Inellipse from triangles to convex quadrilaterals, defining the Lemoine Inellipse of convex quadrilaterals and exploring its properties. The research yielded the following key findings: First, we defined the slice quadrilateral of convex quadrilaterals and investigated its properties. Second, we established the conditions for the existence of the Lemoine Inellipse in convex quadrilaterals. Third, we elucidated the necessary and sufficient conditions for the Lemoine Inellipse of a convex quadrilateral to be a circle. Fourth, we derived the length of the minor axis of the Lemoine Inellipse of a convex quadrilateral. We anticipate that activities such as extending mathematical concepts to broader domains, as demonstrated in this study, will contribute to the advancement of mathematics.