A Study on the Inscribed Prolate Spheroid and Steiner Inscribed Prolate Spheroid of Rectangular Prism

This study extends the concept of the inscribed Prolate Spheroid defined of tetrahedron to explore the inscribed Prolate Spheroid of Rectangular prism. Furthermore, it extends the concept of the Steiner inellipse, defined in triangles andquadrilaterals on a plane, to three dimensions to explore the Steiner inscribed Prolate Spheroid of Rectangular prism. Through this study, the following results were obtained. First, the positional relationship of the two foci of the inscribedProlate Spheroid of Rectangular prism was elucidated. Second, the existence and number of inscribed Prolate Spheroid of Rectangular prism were determined. Third, the positions of the two foci of the Steiner inscribed Prolate Spheroid ofRectangular prism were clarified. Fourth, the necessary and sufficient conditions for the existence of the Steiner inscribed Prolate Spheroid of Rectangular prism were discovered. Fifth, the volume ratio between the Rectangular prism and its Steiner inscribed Prolate Spheroid were revealed. Activities like this study, which extend existing mathematical concepts to other domains and apply planar results to three-dimensional figures, are expected to contribute to the advancement of mathematics.