what is the barycenter of a wire tetrahedron?

Question

The barycenter of a wire triangle is the "Spieker" point, namely the center of the inscribed circle of its medial triangle, that is the triangle whose vertices are the midpoints of the sides. Is there a nice caracterisation for the barycenter of a wire tetrahedron? It is easy to prove that it is on the line which joints the spieker point of any face to the barycenter of the triangle formed by the midpoints of the three remaining sides of the tetrahedron and that it divides that line in the ratio of the total weight of the triangle to the total weight of the three remaining sides. But, I cannot do anything with that. I can also prove that if masses are placed at the vertices proportional to the area of the opposite facet, then the barycenter is the center of the inscribed sphere. But...no farther.