2.1. The segment $A B$ is divided into three equal parts by points $C(3,4)$ and $D(5,6)$. Find the coordinates of points $A$ and $B$.
Solution. Let the coordinates of points $A$ and $B$ be: $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$. For segment $A D$, point $C$ is the midpoint, so $\lambda=A C / C D=1$ and by the formulas for dividing a segment in a given ratio (2.6), we get

$$
x_{C}=\frac{x_{1}+\lambda x_{D}}{1+\lambda}=\frac{x_{1}+x_{D}}{2}, y_{C}=\frac{y_{1}+\lambda y_{D}}{1+\lambda}=\frac{y_{1}+y_{D}}{2} \text {. }
$$

Substitute the coordinates $x_{C}, y_{C}$, $x_{D}, y_{D}$ into the last equalities:

$$
3=\left(x_{1}+5\right) / 2,4=\left(y_{1}+6\right) / 2
$$

from which we find $x_{1}=1, y_{1}=2$. Point $A$ has coordinates $A(1,2)$.

Since point $D$ is the midpoint of segment $C B$, then $x_{D}=$ $=\left(x_{C}+x_{2}\right) / 2$, or $5=\left(3+x_{2}\right) / 2$, from which $x_{2}=7$.

$$
y_{D}=\left(y_{C}+y_{2}\right) / 2,6=\left(4+y_{2}\right) / 2
$$

from which $y_{2}=8$. We obtained $B(7,8)$.

Answer: $A(1,2), B(7,8)$.