We can intuitively look at the picture: the topic is to prove AB+BD=AC, which is equivalent to EB+BD=AC, that is, ED=AC, and you have proved AE=AC, then we only need to prove AE=ED, that is, to go in the direction of angle EAD= angle EDA.

This is also a good proof that, with the sum of external angles, the base angles of isosceles triangles are equal, and the bisector AD:

Angle EAD= Angle EAB+ Angle BAD= 1/2 Angle ABD+ Angle BAD+ Angle C+ Angle DAC= Angle BDA.

Do you understand?