3.20. In the planes of the faces of the tetrahedron \(ABCD\), points \(A_1, B_1, C_1\), and \(D_1\) are taken such that the lines \(AA_1, BB_1, CC_1\), and \(DD_1\) are parallel. Find the ratio of the volumes of the tetrahedra \(ABCD\) and \(A_1B_1C_1D_1\).

## § 4. Calculation of Volume
3.20. Let $A_{2}$ be the intersection point of the line $A A_{1}$ with the plane $B_{1} C_{1} D_{1}$. We will prove that $A_{1} A_{2} = 3 A_{1} A$. Then $V_{A B C D} : V_{A_{2} B C D} = 1: 3$; using the result of problem 3.19, we finally get $V_{A B C D} : V_{A_{1} B_{1} C_{1} D_{1}} = V_{A B C D} : V_{A_{2} B \sim D} = 1: 3$.

Among the collinear vectors $\overrightarrow{B B}_{1}, \overrightarrow{C C}_{1}$, and $\overrightarrow{D D}_{1}$, there will be two that are in the same direction; for definiteness, let's assume that the vectors $\overrightarrow{B B}_{1}$ and $\overrightarrow{C C}_{1}$ are in the same direction. Let $M$ be the intersection point of the lines $B C_{1}$ and $C B_{1}$. The lines $B C_{1}$ and $C B_{1}$ belong to the planes $A D B$ and $A D C$ respectively, so the point $M$ belongs to the line $A D$. Draw a plane through the parallel lines $A A_{1}$ and $D D_{1}$; it passes through the point $M$ and intersects the segments $B C$ and $B_{1} C_{1}$ at some points $L$ and $K$ (Fig. 28). It is easy to verify that $M$ is the midpoint of the segment $K L$, the point $A$ belongs to the lines $D M$ and $D_{1} L$, the point $A_{1}$ to the line $D L$, and the point $A_{2}$ to the line $D_{1} K$. Therefore, $\overrightarrow{A_{1} A} : \overrightarrow{A A}_{2} = \overrightarrow{L M} : \overrightarrow{L K} = 1: 2$, which means $A_{1} A_{2} = 3 A A_{1}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_3f2bc5f5a1af26514cfag-056.jpg?height=480&width=538&top_left_y=828&top_left_x=305)

Fig. 28