This problem has made me feel quite embarrassed.

AB, FE and DC are perpendiculars to BC.

FE is 2 and AB is 6. We are looking for DC.

I believe the solution might come using the similarity of some triangles.

I have found that ABC and ECF are similar, hence BC=3EC.

And BFA is similar to CFD, but I couldn't make a great use of this.

I suppose I need to find a relation between BC and the triangle CDF... in case there is any, that is...

If you have a different solution that does not make use of similar triangles, it is, of course, as much appreciated.