To solve the problem, let's analyze the figure and equations given. The triangle is divided into segments by a line $\overline{ED}$ that is parallel to $\overline{BC}$. This means we can apply the Basic Proportionality Theorem (Thales' Theorem), which states that:

If a line is drawn parallel to one side of a triangle and intersects the other two sides, then it divides the two sides in the same ratio.

Thus, for triangle $\triangle ABC$, with $\overline{ED} \parallel \overline{BC}$, the following proportionality relationships hold true: $\frac{AD}{DB} = \frac{AE}{EC}.$

Let us evaluate each statement:

1. $\frac{AD}{DB} = \frac{AE}{EC}$

This is directly based on the Basic Proportionality Theorem, so True.

2. $\frac{DB}{AB} = \frac{AE}{AC}$

Here, $AB = AD + DB$ and $AC = AE + EC$. Dividing each term as $\frac{DB}{AB}$ and $\frac{AE}{AC}$, the relationship follows proportionality. Hence, True.

3. $\frac{AD}{AE} = \frac{DB}{EC}$

Using the proportionality rule, the ratios $\frac{AD}{DB}$ and $\frac{AE}{EC}$ are equal. Rewriting: $\frac{AD}{AE} = \frac{DB}{EC}.$ Thus, this statement is True.

4. $\frac{DB}{AE} = \frac{AD}{EC}$

Rewriting the relationship $\frac{AD}{DB} = \frac{AE}{EC}$ does not imply that $\frac{DB}{AE} = \frac{AD}{EC}$. This is a mismatch. Hence, False.

5. $\frac{AD}{AB} = \frac{AE}{AC}$

Using proportionality and the segment definitions ($AB = AD + DB$, $AC = AE + EC$), we find this statement holds. Hence, True.

6. $\frac{AD}{AB} = \frac{EC}{AC}$

This is incorrect because $\frac{AD}{AB}$ should correspond to $\frac{AE}{AC}$, not $\frac{EC}{AC}$. Hence, False.

Final Answers:

1. True
2. True
3. True
4. False
5. True
6. False

Would you like further clarification or detailed steps?

Here are 5 questions related to this topic:

1. What is the Basic Proportionality Theorem, and how is it applied?
2. How can the proportionality be proven using similar triangles?
3. Why does parallelism between $\overline{ED}$ and $\overline{BC}$ matter?
4. How do we derive segment proportions from larger triangles?
5. Can proportionality hold if the dividing line is not parallel?

Tip: Always verify whether given lines are parallel before applying the Basic Proportionality Theorem.