Generalizations
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. (Landy. See Wernicke for an earlier result.)
Routh's theorem gives the area of the triangle formed by three cevians in the case that they are not concurrent. Ceva's theorem can be obtained from it by setting the area equal to zero and solving.
The analogue of the theorem for general polygons in the plane has been known since the early nineteenth century (Grünbaum & Shephard 1995, p. 266). The theorem has also been generalized to triangles on other surfaces of constant curvature (Masal'tsev 1994).
Read more about this topic: Ceva's Theorem