2. In the plane, three points $A^{\prime}, B^{\prime}$, and $C^{\prime}$ are given. Determine three points $A, B, C$ in the plane such that $A$ is the midpoint of $C C^{\prime}$, $B$ is the midpoint of $A A^{\prime}$, and $C$ is the midpoint of $B B^{\prime}$.
Solution. In the plane, we set up a rectangular coordinate system $x O y$. Let the coordinates of the given points in this system be: $A^{\prime}=\left(a^{\prime}, a^{\prime \prime}\right), B^{\prime}=\left(b^{\prime}, b^{\prime \prime}\right)$ and $C^{\prime}=\left(c^{\prime}, c^{\prime}\right)$. Let the coordinates of the desired points be $A\left(a_{1}, a_{2}\right), B\left(b_{1}, b_{2}\right)$ and $C\left(c_{1}, c_{2}\right)$. From the conditions for bisecting, we get:

$$
\left(a_{1}, a_{2}\right)=\left(\frac{c^{\prime}+c_{1}}{2}, \frac{c^{\prime \prime}+c_{2}}{2}\right),\left(b_{1}, b_{2}\right)=\left(\frac{a^{\prime}+a_{1}}{2}, \frac{a^{\prime \prime}+a_{2}}{2}\right),\left(c_{1}, c_{2}\right)=\left(\frac{b^{\prime}+b_{1}}{2}, \frac{b^{\prime \prime}+b_{2}}{2}\right) .
$$

By equating the corresponding coordinates and solving the resulting system of linear equations, we get

$$
\begin{array}{ll}
a_{1}=\frac{a^{\prime}+2 b^{\prime}+4 c^{\prime}}{7}, & a_{2}=\frac{a^{\prime \prime}+2 b^{\prime \prime}+4 c^{\prime \prime}}{7} \\
b_{1}=\frac{b^{\prime}+2 c^{\prime}+4 a^{\prime}}{7}, & b_{2}=\frac{b^{\prime \prime}+2 c^{\prime \prime}+4 a a^{\prime}}{7} \\
c_{1}=\frac{c^{\prime}+2 a^{\prime}+4 b^{\prime}}{7}, & c_{2}=\frac{c^{\prime \prime}+2 a^{\prime \prime}+4 b^{\prime \prime}}{7}
\end{array}
$$

For the solution to be complete, we need to prove that the points $A, B$ and $C$ with such coordinates can be constructed if the coordinates of $A^{\prime}, B^{\prime}$ and $C^{\prime}$ are known (the problem does not ask for the coordinates of $A, B, C$, but rather for these points to be constructed). However, since the construction of a segment with double the length, a segment with the length equal to the sum of the lengths of given segments, and the division of a given segment into 7 parts are known, the points $A, B$ and $C$ can be constructed using the coordinates of $A^{\prime}, B^{\prime}$ and $C^{\prime}$ in the established coordinate system.