Geometry with parallelogram

Question

My question is...
I want to know your another solution.
or I want to know if my solution is appropriate.
and I’d appreciate some feedback on my work.

Mentioned the word)
Parallelogram ABCD, $\angle BAE = \angle CAE$ , $\overline{BE}+\overline{BC}=\overline{BD}$ , Find the $\overline{BC} : \overline{BD}$

Answer 1

Your solution is correct. And I think this is the only geometry way to do it. (You can try using complex plane, but I tried and it's messy). Well done :)

Answer 2

I finished the final conclusion in a different way.(I used circle)
Additionally, Let's solve the problem without the auxiliary line.