To prove that \( BI = CI \) in the given triangle \( \triangle ABC \) where \( AB = AC \) and \( AD \perp BC \), we can follow these steps:

1. **Identify Points**: We have points \( E \) on \( AB \) and \( F \) on \( AC \) such that \( AE = AF \).

2. **Use Isosceles Triangle Properties**: Since \( \triangle ABC \) is isosceles with \( AB = AC \), this means that angles \( \angle ABC \) and \( \angle ACB \) are equal. Denote these angles as \( \alpha \).

3. **Point D on BC**: Since \( AD \perp BC \), triangles \( \triangle ABD \) and \( \triangle ACD \) can be analyzed. In these right triangles, we can apply trigonometric ratios.

4. **Segment Relations**: The segments \( BI \) and \( CI \) can be related through angle properties and congruence due to the isosceles nature of \( \triangle ABC \).

5. **Use the lengths**: Since \( AE = AF \), and \( AD \) is the height from \( A \) to \( BC \), we can derive relationships involving the segments based on equal lengths and the common height lined from point \( A \).

6. **Conclusion**: By symmetry in triangle \( ABC \), the lengths \( BI \) and \( CI \) must be equal because the setup is symmetrical about the altitude \( AD \).

Thus, we conclude \( BI = CI \).

Let me know if you need further details or specific references to triangle properties!