4. As shown in Figure 1, point $C$ is any point on segment $A B$, $\triangle D A C$ and $\triangle E C B$ are both equilateral triangles, connect $A E$ intersecting $C D$ at point $M$, connect $B D$ intersecting $C E$ at point $N$. When point $C$ moves on segment $A B$, among the following 4 equations:
$$
\begin{array}{l}
A E=B D, C M=C N, \\
A M=D N, B N=E M
\end{array}
$$

the number of equations that always hold is ( ) .
(A) 1
(B) 2
(C) 3
(D) 4
4.D.

As shown in Figure 1, by rotating $\triangle C B D$ counterclockwise around point $C$ by $60^{\circ}$, we obtain $\triangle C E A$. Thus,
$$
\begin{array}{l}
A E=D B, \text { and } \angle C A E=\angle C D B . \\
\text { Also, } \angle A C M=\angle D C N=60^{\circ}, A C=D C, \text { hence } \\
\triangle A C M \cong \triangle D C N .
\end{array}
$$

Therefore, $C M=C N, A M=D N$.
Similarly, $\triangle B C N \cong \triangle E C M$, so $B N=E M$.