On the Straightedge and Compass Construction of the Foci of an Inscribed Ellipse with a Given Center

In this paper, we present a novel and elegant geometric construction for determining thefoci of an inscribed ellipse (inellipse) within a reference triangle ABC, when only the center Oof the ellipse is known. By leveraging the projective properties of the triangle’s centroid andthe isotomic conjugate of the Brianchon point, we first establish a direct method to locate thepoints of tangency. Subsequently, through a series of lemmas involving symmedian lines andcyclic quadrilaterals, we uniquely determine the positions of the foci using only a straightedgeand compass. Finally, it should be noted that the specific and prominent case of the Steinerinellipse has been comprehensively analyzed by the author in [1].