Make BC's vertical line pass through point D and intersect BC at point G.

You can know the length of BG and thus the length of BD.

According to 1, the triangle BAE is similar to the triangle CBD, and the similarity ratio can BE obtained according to the line length, so that the lengths of BD and BE can be obtained.

It can also be proved that the triangle BFE is similar to the triangle DFA, and the similarity ratio can be obtained.

And BF+FD=BD.

Then find the length of BF.