Final answer:
The question requires proving that triangle ADE is isosceles and calculating angle ADB using geometric theorems and postulates, but without a figure or additional information, a definitive solution cannot be provided.
Step-by-step explanation:
The question delves into the realm of geometry, where we need to prove that triangle ADE is isosceles and calculate the size of angle ADB based on the given information.
Since AB = AC and BD = EC are given, we use the properties of isosceles triangles and congruent segments to deduce the necessary angles and sides. For angle DAE, which is given as 40°, we can infer certain properties based on isosceles triangles and their angles.
However, without a figure to reference or additional information, it's not possible to provide an accurate proof or calculation for angle ADB. The necessary steps would typically involve using congruence and similarity postulates, angle sum properties, and potentially the Law of Sines or Cosines, depending on the construction of the figure.