15.5. (GDR, 74). Determine into how many parts a regular tetrahedron is divided by six planes, each of which passes through one edge and the midpoint of the opposite edge of the tetrahedron. Find the volume of each of the parts if the volume of the tetrahedron is 1.
15.5. Since any of the planes drawn contains one of the segments connecting the midpoints of opposite edges of the tetrahedron, all planes pass through the point of intersection of these segments, which lies inside the tetrahedron (see Theorem 92). Therefore, the six drawn planes divide the entire space into polyhedral angles with a common vertex, and thus any part of the tetrahedron has at least one face belonging to the face of the tetrahedron. On the other hand, no two faces of the tetrahedron can serve as the boundary of the same part, since any two of its faces are separated from each other by a plane passing through their common edge. Thus, the number of parts of the tetrahedron is equal to the number of parts into which its surface is divided, and since each face of the tetrahedron is divided into 6 parts (medians are drawn in it), there are a total of $6 \times 4 = 24$ parts. From considerations of symmetry (with respect to any of the six planes), it follows that all parts of the tetrahedron are equal to each other, and the volume of each of them is $1 / 24$.