In a tetrahedron, we draw a plane through the midpoint of each edge and the midpoint of the opposite edge. How many parts do these planes divide the tetrahedron into, and how do the volumes of these parts compare to the volume of the tetrahedron?
Let the vertices of the tetrahedron be denoted by \( A_{1}, A_{2}, A_{3}, A_{4} \), the centroid by \( S \), the midpoint of the edge \( A_{1} A_{3} \) by \( F_{13} \), and the midpoint of the edge \( A_{2} A_{4} \) by \( F_{24} \). As is well known, the medians of a tetrahedron are the lines connecting a vertex to the centroid of the opposite triangular face. We also know that the medians of a tetrahedron intersect at the point that is one-fourth of the way from the vertex to the opposite face, which is called the centroid of the tetrahedron. (L. II. textbook, page 154.)

![](https://cdn.mathpix.com/cropped/2024_05_02_0d4f923dd92d9d80601fg-1.jpg?height=621&width=819&top_left_y=351&top_left_x=642)

In the following, we will prove that each of the planes we consider passes through the centroid of the tetrahedron.

Consider, for example, the planes passing through the points \( A_{2}, A_{4}, F_{13} \) and \( A_{1}, A_{3}, F_{24} \). Since the lines \( A_{1} A_{3} \) and \( A_{2} A_{4} \) are not in the same plane, the planes do not coincide, so they have a line of intersection, which is precisely the line passing through the points \( F_{13} \) and \( F_{24} \). Both the plane \( A_{2} A_{4} F_{13} \) and the plane \( A_{1} A_{3} F_{24} \) contain the centroid of the tetrahedron, so \( S \) lies on the line of intersection of the two planes; this proves our statement.

If we connect the centroid of the tetrahedron to its vertices, the resulting 4 smaller tetrahedra each have a volume that is one-fourth of the original volume, because their base is a face of the tetrahedron, and their height is one-fourth of the corresponding height of the tetrahedron. The planes mentioned in the problem further divide these smaller tetrahedra into 6 parts each, since the planes pass through the centroid of the tetrahedron and intersect two faces of the tetrahedron along medians. Thus, a total of \( 6 \times 4 = 24 \) parts are formed. The volumes of these parts are all equal, because the medians of a triangle divide it into 6 equal-area parts, and these will be the bases of the smaller tetrahedra. Therefore, the volume of one part is one twenty-fourth of the volume of the tetrahedron.