3.31. Tetrahedra $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$ are positioned such that all vertices of each lie in the corresponding planes of the faces of the other tetrahedron ( $A$ lies in the plane $B_{1} C_{1} D_{1}$, etc.). Moreover, $A_{1}$ coincides with the centroid of triangle $B C D$, and the lines $B D_{1}, C B_{1}$, and $D C_{1}$ bisect the segments $A C, A D$, and $A B$ respectively. Find the volume of the common part of the tetrahedra, if the volume of tetrahedron $A B C D$ is $V$.

## 8 5. Auxiliary volume



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3.31. Let $K, L$ and $M$ be the midpoints of segments $AB, AC$ and $AD$. First, we prove that $K$ is the midpoint of segment $DC_1$. Point $B$ lies in the plane $A_1C_1D_1$, therefore point $C_1$ lies in the plane $A_1LB$. We extend the tetrahedron $ABCD$ to a triangular prism by adding vertices $S$ and $T$, where $\overrightarrow{AS}=\overrightarrow{DB}$ and $\overrightarrow{AT}=\overrightarrow{DC}$. The plane $A_1LB$ passes through the midpoints of sides $CD$ and $AT$ of the parallelogram $CDAT$, therefore it contains the line $BS$. Consequently, $S$ is the intersection point of the line $DK$ with the plane $A_1LB$, i.e., $S=C_1$. Similarly, it can be proven that $L$ and $M$ are the midpoints of segments $BD_1$ and $CB_1$. Thus, the tetrahedron $A_1B_1C_1D_1$ is bounded by the plane $B_1C_1D_1$, passing through point $A$ parallel to the face $BCD$, and the planes

![](https://cdn.mathpix.com/cropped/2024_05_21_3f2bc5f5a1af26514cfag-060.jpg?height=505&width=524&top_left_y=959&top_left_x=518)

Fig. 31 planes $A_1LB, A_1MC$ and $A_1KD$.

Let $Q$ be the midpoint of $BC$, and $P$ be the intersection point of $BL$ and $KQ$ (Fig. 31). The plane $A_1KD$ cuts off a tetrahedron $DKBQ$ from the tetrahedron $ABCD$, the volume of which is $V/4$; the planes $A_1LB$ and $A_1MC$ cut off tetrahedrons of the same volume. For example, for the tetrahedrons cut off by the planes $A_1KD$ and $A_1LB$, the common tetrahedron $A_1BPQ$ has a volume of $V/24$. Therefore, the volume of the common part of the tetrahedrons $ABCD$ and $A_1B_1C_1D_1$ is equal to $V(1-3/4+3/24)=3V/8$.