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!

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.