6. 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, and they are on the same side of $A B$. Connect $A E$ to intersect $C D$ at point $M$, and connect $B D$ to intersect $C E$ at point $N$.
The following equations are obtained:
(1) $A E=B D$,
(2) $C M=C N$,
(3) $A M=D N$,
(4) $B N=E M$.
When $\triangle D A C$ rotates around point $C$, the number of the above 4 equations that always hold is ( ).
(A) exactly 1
(B) exactly 2
(C) exactly 3
(D) all 4

When $\triangle D A C$ rotates around point $C$, the number of the above 4 equations that always hold is ( ).
(A) exactly 1
(B) exactly 2
(C) exactly 3
(D) all 4
6. A.

Since $\triangle A C E$ can be obtained by rotating $\triangle D C B$ clockwise around point $C$ by $60^{\circ}$, we have
$$
A E=B D, \angle C A M=\angle C D N .
$$

From $A C=D C, \angle A C M=\angle D C N=60^{\circ}$, we get $\triangle A C M \cong \triangle D C N$.
Thus, all four equations hold.
However, when $\triangle D A C$ rotates around point $C$, the equation $\angle A C M = \angle D C N = 60^{\circ}$ is disrupted, and consequently, $\triangle A C M \cong \triangle D C N$ is also disrupted. Therefore, only $A E=B D$ always holds. Note: $A E=B D$ is a substantial invariant under the motion, while the other three equations depend on the condition $\angle A C M=\angle D C N=60^{\circ}$.