10. Let $A_{1}, A_{2}, A_{3}, A_{4}$ be pairwise distinct points in a Cartesian coordinate system. If
$$
\begin{array}{l}
\overrightarrow{A_{1} A_{3}}=\lambda \overrightarrow{A_{1} A_{2}}(\lambda \in \mathbf{R}), \\
\overrightarrow{A_{1} A_{4}}=\mu \overrightarrow{A_{1} A_{2}}(\mu \in \mathbf{R}),
\end{array}
$$

and $\frac{1}{\lambda}+\frac{1}{\mu}=2$, then the points $A_{3}, A_{4}$ are said to harmonically divide the points $A_{1}, A_{2}$. Given that points $C, D$ harmonically divide points $A, B$, which of the following statements is correct? ( ).
(A) $C$ could be the midpoint of segment $A B$
(B) $D$ could be the midpoint of segment $A B$
(C) Points $C, D$ could simultaneously lie on segment $A B$
(D) Points $C, D$ cannot simultaneously lie on the extension of segment $A B$
10. D.
According to the problem, $\frac{1}{c}+\frac{1}{d}=2$.
If $C$ or $D$ is the midpoint of segment $A B$, then $c=\frac{1}{2}$ or $d=\frac{1}{2}$,

which is a contradiction.
If points $C$ and $D$ are both on segment $A B$, then $0<\frac{1}{c}+\frac{1}{d}<2$,

which is a contradiction.
If points $C$ and $D$ are both on the extension of segment $A B$, then $c>1, d>1 \Rightarrow 0<\frac{1}{c}+\frac{1}{d}<2$.
Therefore, points $C$ and $D$ cannot both be on the extension of segment $A B$.