This study originated from a well-known tool in mathematical problem solving, the “incenter/excenter lemma”, which states that in a triangle, the circumcenters of the three subtriangles formed by the incenter and the vertices lie on the circumcircle of the original triangle. We modified the condition by replacing the incenter with the orthocenter and the circumcenter, and subsequently investigated the center-connecting triangle. Surprisingly, we revealed a subtle and elegant relationship between the orthocentric and circumcentric configurations, which was essentially interpreted as the concurrency of three circles. We further generalized the circumcenter and orthocenter to arbitrary isogonal conjugate points, and, by employing inversion, proved that seven circles concurred at a single point, which always lay on the circumcircle of the reference triangle. By interchanging center lines and common chords, we established geometric relations among four center-connecting triangles—specifically, their similarity and perspectivity. Moreover, we demonstrated that the six vertices of the two perspective triangles were concyclic. We then proceeded to investigate the geometric entities associated with perspectivity—the center of perspectivity and the axis of perspectivity—deriving elegant results using tools from projective geometry.