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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.

As shown in the figure below, AB and CD are diameters of circle O. Add AE through A such that AE || CD and AE intersects circle O at E. Connect BD, DE. Prove: BD = DE.

Explore the proof that BD = DE in a circle where AB and CD are diameters, and AE is drawn parallel to CD. Learn about the geometric properties of circles, diameters, and congruent triangles, with step-by-step explanations. Suitable for students in Grades 9-11.

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