https://file.aiquickdraw.com/mathbot/file/173016133210348whj9au.jpg

Let's examine the problem.

### Given Information:
- \( \angle AOB = \angle AOC \)
- \( AB \cong AC \) (segments \( AB \) and \( AC \) are equal in length)

### Conclusion to Prove:
- \( AD \) is the perpendicular bisector of \( BC \).
- Hint: Show that points \( A \) and \( O \) are equidistant from points \( B \) and \( C \).

### Solution Outline:
1. **Equal Segments**:
   Since \( AB \cong AC \), triangle \( \triangle ABC \) is isosceles with \( AB = AC \).
   
2. **Equal Angles**:
   Since \( \angle AOB = \angle AOC \), \( OB = OC \), indicating that \( O \) lies on the perpendicular bisector of \( BC \).
   
3. **Equidistant Points**:
   The equidistance of \( A \) and \( O \) from points \( B \) and \( C \) implies that \( AD \) must bisect \( BC \) at \( D \) and be perpendicular to \( BC \).

4. **Conclusion**:
   By showing that \( AD \) bisects \( BC \) and is perpendicular to it, we conclude that \( AD \) is the perpendicular bisector of \( BC \).

Would you like further details on any part of this proof or have any questions?

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### Related Questions:
1. What is the significance of isosceles triangles in geometric proofs?
2. How does the property of equidistant points help in constructing perpendicular bisectors?
3. Can you explain why an isosceles triangle's altitude is also its perpendicular bisector?
4. What additional properties does the center of a circle provide in this configuration?
5. How can we prove that two angles are equal based on segment congruence?

### Tip:
Remember, for any triangle with two equal sides, the altitude from the vertex angle not only bisects the base but is also perpendicular to it.

Given that O is the center of the circle, AB is congruent to AC, and angle AOB is congruent to angle AOC, prove that AD is the perpendicular bisector of BC.

This problem demonstrates how to prove that AD is the perpendicular bisector of BC given a circle with center O, where AB is congruent to AC. The proof involves understanding key geometric properties such as congruent angles, perpendicular bisectors, and equidistant points in circles. Suitable for Grades 9-10.

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