This research stems from an extension of a problem involving the centroid of a triangle, aiming to investigate the properties of a dynamic four-center n-gon constructed from a given n-gon on the plane and a moving point 𝑃. The four centers considered are the centroid, orthocenter, circumcenter, and incenter. The study examines the geometric relationships and proportional characteristics between the original n-gon and the corresponding dynamic four-center n-gon, the trajectories of the vertices of the dynamic four-center n-gon as the point 𝑃 traverses the circumcircle of the original n-gon. Finally, this research extends to the geometric properties arising from iterative constructions of such four-center n-gons.