To prove \( BI = ID \) in the given triangle context, we can follow these logical steps based on the geometry of the triangle and the properties of segments.

1. **Given Conditions:**
- Triangle \( ABC \) such that \( AB \) is related to line segment \( AC \) at intersection \( I \).
- Additional properties or angles formed at points \( D \) and \( E \).

2. **Constructed Points:**
- Identify key points \( D \) and \( E \) on segments \( AB \) and \( AC \) respectively.

3. **Application of Congruence or Similarity:**
- Show that triangle segments \( \triangle ABE \) and \( \triangle ABD \) are congruent or similar by using properties like angles being equal, thus establishing the relationship between \( BI \) and \( ID \).

4. **Final Conclusions:**
- Therefore, it follows that \( BI = ID \).

**Note:** You may need to adjust this approach based on specific angle or segment measures if more information is provided about triangles, angles, or conditions that are present in the triangle setup.