In the prism $A B C A 1 B 1 C 1$, the medians of the bases $A B C$ and $A 1 B 1 C 1$ intersect at points $O$ and $O 1$, respectively. A line is drawn through the midpoint of segment $O O 1$, parallel to the line $C A 1$. Find the length of the segment of this line that lies inside the prism, if $C A 1=a$.
Let $BM$ and $B1M1$ be the medians of triangles $ABC$ and $A1B1C1$, $P$ be the midpoint of $OO1$, $F$ be the midpoint of $A1C$, and the line $FP$ intersects the edge $BB1$ at point $D$. We draw a plane $\alpha$ through the line $A1C$ and the point $P$ not lying on it. Since points $P$ and $F$ lie in the plane $\alpha$, point $D$ also lies in it. Therefore, the section of the given prism by the plane $\alpha$ is the triangle $A1DC$. Draw a line through point $P$ in the plane $\alpha$ parallel to $A1C$. Let it intersect the segments $A1D$ and $CD$ at points $K$ and $N$. Then $KN$ is the desired segment and

$$
\frac{KN}{A1C} = \frac{DP}{DF} = \frac{BO}{BM} = \frac{2}{3}
$$

Therefore,

$$
KN = \frac{2}{3} A1C = \frac{2}{3} a.
$$

## Answer

$\frac{2}{3} a$.