The Maximum of the Volume of a Cevian Simplex and its Parts

The Cevian triangle corresponding to an interior point M of a triangle is the triangle determined by the feet of the three Cevians concurrent at M. It is known that the area of the Cevian triangle for an interior point M of a triangle is at most 14 of the area of the triangle, with maximum attained when M is the triangle’s centroid. This can be generalized from triangles to n-dimensional simplices, with 14 replaced by 1nn, using barycentric coordinates. We also use this method to solve two optimization problems about the parts of this simplex.