Let's break down the problem step-by-step to prove that $BD = DE$.

Given:

• $\overline{AB}$ and $\overline{CD}$ are diameters of circle $O$, meaning both pass through the center of the circle.
• Line $\overline{AE}$ is drawn parallel to $\overline{CD}$ and intersects the circle at $E$.
• You need to prove that $BD = DE$.

Approach:

1. Since $\overline{AB}$ and $\overline{CD}$ are diameters, they both pass through the center $O$ and are perpendicular to each other (diameters of a circle form a right angle at the center).

2. $\overline{AE} \parallel \overline{CD}$ indicates that the quadrilateral $BODE$ is symmetric in some way.

3. Notice that triangles $\triangle BOD$ and $\triangle OED$ both share common angles:

   • $\angle BOD$ is a right angle, as $\overline{AB}$ and $\overline{CD}$ are perpendicular diameters.
   • $\triangle BOD$ and $\triangle OED$ are also isosceles triangles because $O$ is the center of the circle, so $OB = OD = OE$ (since all are radii of the circle).

4. The fact that $\overline{AE} \parallel \overline{CD}$ makes triangles $\triangle OBD$ and $\triangle OED$ congruent by the Angle-Angle-Angle (AAA) similarity criterion.

5. Since the triangles $\triangle BOD$ and $\triangle OED$ are congruent, it follows that $BD = DE$, as corresponding sides of congruent triangles are equal.

Conclusion:

By using the properties of the circle (diameters and radii), parallel lines, and congruent triangles, we can prove that $BD = DE$.

Would you like more detailed steps or clarification on any part? Feel free to ask!

Here are 5 related questions to expand on this concept:

1. How does the fact that diameters are perpendicular impact the symmetry of the figure?
2. Can you always conclude that triangles are congruent when two sides are equal in a circle?
3. Why is $AE \parallel CD$ important for proving the equality of the segments?
4. How would the proof change if $AE$ were not parallel to $CD$?
5. What other properties of circles are useful for proving equal line segments?

Tip: In circle geometry, always look for symmetry and properties of diameters and radii, as they often simplify proofs.