Given that \( a \) represents the measure of \( \angle EAB \), we can proceed with a trigonometric approach to find the ratio \( \frac{DE}{CE} \).

### Problem Recap:
- \( ABCD \) is a trapezium with \( AD \parallel BC \).
- \( E \) is the midpoint of \( AD \).
- \( \angle ABE = \angle BCE = 90^\circ \).
- \( \angle EAB = a \), and we need to find \( \frac{DE}{CE} \).

### Solution with \( a = \angle EAB \):

1. **Label the Sides Using Trigonometry:**
   - Let \( AE = \frac{b}{2} \), since \( E \) is the midpoint of \( AD \).
   - From the definition of cosine and sine in triangle \( \triangle ABE \):
     \[
     AB = AE \cos(a) = \frac{b}{2} \cos(a)
     \]
     \[
     BE = AE \sin(a) = \frac{b}{2} \sin(a)
     \]

2. **Consider Triangle \( \triangle BCE \):**
   - Since \( \triangle ABE \sim \triangle BCE \), the corresponding sides are proportional:
     \[
     \frac{AB}{BC} = \frac{BE}{CE}
     \]
   - Here, \( AB = \frac{b}{2} \cos(a) \) and \( BC = a \).

3. **Express \( CE \):**
   \[
   \frac{\frac{b}{2} \cos(a)}{a} = \frac{\frac{b}{2} \sin(a)}{CE}
   \]
   Simplifying for \( CE \):
   \[
   CE = \frac{a \sin(a)}{\cos(a)}
   \]

4. **Find \( DE \):**
   - \( DE = \frac{b}{2} \), as \( E \) is the midpoint of \( AD \).

5. **Compute the Ratio \( \frac{DE}{CE} \):**
   \[
   \frac{DE}{CE} = \frac{\frac{b}{2}}{\frac{a \sin(a)}{\cos(a)}} = \frac{b \cos(a)}{2a \sin(a)}
   \]

Thus, the ratio \( \frac{DE}{CE} \) in terms of \( a \) is:
\[
\boxed{\frac{DE}{CE} = \frac{b \cos(a)}{2a \sin(a)}}
\]

### Do you need any further clarification or a different aspect of the problem?

#### Related Questions:
1. How does this ratio change if \( a \) approaches \( 0^\circ \) or \( 90^\circ \)?
2. Can you derive an expression for \( b \) in terms of \( a \) and other sides?
3. How would the relationship change if \( E \) was not the midpoint of \( AD \)?
4. What is the implication of this ratio in terms of the area of the trapezium?
5. What are the conditions under which this ratio becomes 1?

#### Tip:
In problems involving angles and trigonometric functions, it's crucial to express all relevant sides and ratios in terms of the same trigonometric identities to simplify the solution process effectively.

Explore the detailed solution for calculating the ratio DE to CE in a trapezium using trigonometric principles. Understand the application of cosine and sine functions in determining side ratios. Suitable for advanced high school students and enthusiasts of geometry and trigonometry.

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