In triangle $ABC$, points $D$ and $F$ are on $\overline{AB},$ and $E$ is on $\overline{AC}$ such that $\overline{DE}\parallel \overline{BC}$ and $\overline{EF}\parallel \overline{CD}$. If $AF = 9$ and $DF = 3$, then what is $BD$?

Answer 1

Taking all that gibberish out of your post, leaves me with:

In triangle ABC, points D and F are on AB and E is on AC such that DE || BC and EF || CD. If AF = 9 and DF = 3, then what is BD?

I cannot reconcile your description into a viable diagram.