14. As shown in Figure 4, in rectangle $A B C D$, $A B = 2 A D$, and $E$ is the midpoint of side $A B$. Fold $\triangle A D E$ along $D E$ to form $\triangle A_{1} D E$. If $M$ is the midpoint of line segment $A_{1} C$, then during the folding process of $\triangle A D E$, which of the following statements is correct? 
(1) The length of line segment $B M$ is a constant;
(2) There exists a position such that $D E \perp A_{1} C$;
(3) The path of point $M$ is a circle;
(4) There exists a position such that $M B \perp$ plane $A_{1} D E$.
14. (1)(3).

Take the midpoint of $CD$ as $F$. Then
$M F / / D A, B F / / D E$.
Thus, plane $M B F / /$ plane $A_{1} D E$
$\Rightarrow M B / /$ plane $A_{1} D E \Rightarrow$ statement (4) is incorrect.
Since $\angle M F B = \angle A_{1} D E, M F = \frac{1}{2} A_{1} D, F B = D E$ are all constants, it follows that $M B$ is also a constant. Therefore, statement (1) is correct.

Since $B$ is a fixed point, point $M$ lies on a sphere with $B$ as the center and $M B$ as the radius. Statement (3) is correct.

Since the projection of $A_{1} C$ in the plane $A B C D$ is $A C$, and $A C$ is not perpendicular to $D E$, statement (2) is incorrect.