80. In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the lengths of the edges $A B, B C$, and $B B_{1}$ are equal to $2 a, a$, and $a$, respectively. Point $E$ is the midpoint of edge $B C$. Vertices $M$ and $N$ of a regular tetrahedron $M N P Q$ lie on the line $C_{1} E$, and vertices $P$ and $Q$ lie on the line passing through point $B_{1}$ and intersecting the line $A D$ at point $F$. Find: a) the length of segment $D F$; b) the distance between the midpoints of segments $M N$ and $P Q$.
80. Since opposite edges of a regular tetrahedron are perpendicular, the lines \( C_{1} E \) and \( B_{1} F \) (Fig. 5) must be perpendicular.

![](https://cdn.mathpix.com/cropped/2024_05_21_9b03e22d2b8ca67aa653g-058.jpg?height=289&width=530&top_left_y=716&top_left_x=361)

Fig. 5,

If \( K \) is the midpoint of \( C_{1} C \), then since the lines \( B_{1} K \) and \( B_{1} A_{\mathbf{i}} \) are perpendicular to the line \( C_{1} E \), the line \( B_{1} F \) must lie in the plane passing through \( B_{1} K \) and \( B_{1} A_{1} \), and hence \( A_{1} F \| B_{1} K \) and, therefore, \( |D F| = a \) (this is the answer to part a)).

b) The distance between the midpoints \( M N \) and \( P Q \) is equal to the distance between the lines \( B_{1} F \) and \( C_{1} E \). This can be found by equating different expressions for the volume of the tetrahedron \( F B_{1} C_{1} E \):

\[
\frac{1}{3} S_{B_{1} C_{1} E} 2 a = \frac{1}{6} \left| F B_{1} \right| \cdot \left| C_{1} E \right| \cdot x
\]

From this, \( x = \frac{4 a}{3 \sqrt{5}} \).