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The problem is in Persian and describes a geometry question involving a square \( ABCD \). Here’s a summary in English:

1. In square \( ABCD \):
   - A line segment \( AE \) is drawn from vertex \( A \) such that \( E \) lies on side \( DC \).
   - Point \( F \) is the intersection of line segment \( BC \) with a line that bisects angle \( BAE \).
   
2. The task is to find the sum of the lengths \( BF + DE \).

### Solution Outline
1. **Understand the Square**: Since \( ABCD \) is a square, each side is equal, and angles are \( 90^\circ \).
2. **Angle Bisector**: The line bisecting angle \( BAE \) suggests symmetry or specific angle relationships.
3. **Using Geometry Properties**: Depending on known angles and side lengths, apply geometric properties (like the Pythagorean theorem or trigonometric ratios) to calculate \( BF \) and \( DE \).

Would you like me to proceed with a full solution, or do you have any specific questions about this problem?

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### Further Questions for Deeper Understanding
1. How do we determine the coordinates of points \( E \) and \( F \) in relation to square \( ABCD \)?
2. What geometric properties are helpful in solving problems involving squares?
3. How does an angle bisector affect the lengths of intersecting segments?
4. What role does symmetry play in simplifying geometric calculations within squares?
5. Could this problem be solved using trigonometry, and if so, how?

### Tip
In geometry, always mark known values and angles on a diagram to make visualization easier and help with identifying symmetries or specific properties of the shape.

In square ABCD, a line segment AE is drawn from vertex A, such that E lies on side DC. A line bisects angle BAE, intersecting side BC at point F. Find the sum of the lengths BF + DE.

Explore a detailed solution to a geometry problem involving a square ABCD, where an angle bisector intersects side BC. Learn how to find the sum of BF + DE, using properties of squares and the Angle Bisector Theorem, suitable for students in Grades 9-11.

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