the theorem can be generalized to higher-dimensional simplexes using barycentric coordinates. define a cevian of an "n"-simplex as a ray from each vertex to a point on the opposite ("n"-1)-face (facet). then the cevians are concurrent if and only if a mass distribution can be assigned to the vertices such that each cevian intersects the opposite facet at its center of mass. moreover, the intersection point of the cevians is the center of mass of the simplex.