https://file.aiquickdraw.com/mathbot/file/17265819793093j4qkn41.jpg

The problem provides the following information:

- In triangle \( \Delta ABC \), \( \angle DAE = 90^\circ \),
- \( AD = BD \) and \( AE = EC \),
- We are asked to find \( \angle BAC \).

### Step-by-step approach:

1. **Isosceles triangles:**
   - Since \( AD = BD \), triangle \( \Delta ABD \) is isosceles.
   - Since \( AE = EC \), triangle \( \Delta AEC \) is also isosceles.

2. **Right angle at \( A \):**
   - \( \angle DAE = 90^\circ \) means the line \( DE \) is perpendicular to \( AC \).

3. **Geometrical properties and symmetry:**
   - Since \( AD = BD \) and \( AE = EC \), the two triangles \( \Delta ABD \) and \( \Delta AEC \) are symmetric about \( A \).
   
4. **Total angle at vertex \( A \):**
   - The total angle at \( A \) is composed of the two angles \( \angle BAD \) and \( \angle CAD \), which are equal due to the symmetry of the two isosceles triangles.
   
   Therefore, \( \angle BAC = 90^\circ \), since the triangle symmetry forces each half to be \( 45^\circ \).

### Final Answer:
\( \angle BAC = 90^\circ \).

Let me know if you need any details or clarifications!

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Here are 5 follow-up questions related to this problem:
1. How can we prove the symmetry of the triangles \( \Delta ABD \) and \( \Delta AEC \)?
2. What is the importance of isosceles triangles in geometry?
3. How do perpendicular lines influence the angles in a triangle?
4. Can we generalize this approach to other triangle configurations?
5. How do right triangles help in solving complex geometric problems?

**Tip:** In geometry, identifying equal lengths and angles can simplify complex problems and lead to elegant solutions.

In triangle ABC, ∠DAE = 90°, AD = BD, and AE = EC. Find ∠BAC.

Solve the geometry problem where triangle ABC has AD = BD and AE = EC with a 90° angle at ∠DAE. Learn to find ∠BAC using properties of isosceles triangles and triangle angle sum.

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