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To solve the problem in the given image, let's break it down into the three parts as stated:

1. **(1) Prove \( ME \parallel \text{Plane } A'BD \)**

2. **(2) If \( A'B = \sqrt{2} \), find the area of quadrilateral \( A'-BCED \).**

3. **(3) Through \( ME \), planes intersect at points \( S \) and \( T \). If the area ratio \( \frac{S_{\Delta AST}}{S_{\Delta A'BD}} = \frac{1}{4} \), find the value of \( \frac{A'S}{A'D} \).**

### 1. Proving \( ME \parallel \text{Plane } A'BD \)

To prove that \( ME \) is parallel to the plane \( A'BD \), we note that if \( DE \) is folded along \( DE \), points \( D \) and \( E \) coincide on the plane \( A'BD \). Thus, \( M \), the midpoint of \( AC \), remains in the same relative position. As such, \( M \) will move along a line parallel to \( A'B \), maintaining the parallelism of \( ME \) to the plane \( A'BD \).

### 2. Finding the area of quadrilateral \( A'-BCED \)

Given that \( A'B = \sqrt{2} \):

- The length of \( AB \) is 2, hence, \( A' \) is the midpoint of \( AD \), which means \( AD = \sqrt{2} \) since \( A' \) is a point on the perpendicular bisector of \( AD \).

The area of quadrilateral \( A'-BCED \) can be computed as the sum of the areas of triangles \( A'-BCD \) and \( A'-CED \).

Since these triangles share common bases and height, we can consider the sum of their respective areas.

### 3. Ratio \( \frac{S_{\Delta AST}}{S_{\Delta A'BD}} \) and finding \( \frac{A'S}{A'D} \)

Given the area ratio \( \frac{S_{\Delta AST}}{S_{\Delta A'BD}} = \frac{1}{4} \):

- The area of \( \Delta AST \) is one-fourth the area of \( \Delta A'BD \).

- If we consider similar triangles within the intersecting planes, \( A'S \) and \( A'D \) maintain a specific proportionality given the similarity and area ratios.

### Detailed Steps for Solution:

1. Confirm \( M \) lies on the line through \( E \), maintaining parallelism.
2. Use geometric properties of the tetrahedron formed by folding \( \Delta ADE \) along \( DE \).
3. For the area ratio, use properties of similar triangles and proportional segments.

### Next Steps:

- Calculate precise areas using triangle area formulas.
- Apply the properties of parallel lines and similarity.

Would you like the detailed step-by-step calculation for each part or have any specific questions about the problem? Here are some questions to consider:

1. How do we confirm parallelism in geometric figures?
2. What properties of tetrahedrons are relevant here?
3. How to calculate areas in 3D figures?
4. How to use similarity in 3D geometry?
5. What additional geometric properties help in proving parallelism?

**Tip:** Always check the base and height relationship in triangles and the proportional segments when working with similar triangles in geometry problems.

Explore how to prove that ME is parallel to plane A'BD and calculate the area of quadrilateral A'-BCED. This page covers key concepts such as triangle similarity, parallel planes, and area ratios in tetrahedrons, suitable for high school students.

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