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 $AF = 8$ and $DF = 4$, then what is $BD$?
A
8
F
4
D E
B C
Because EF is parallel to CD, then AF/ DF = AE / EC
So 8 /4 = AE /EC = 2
So AE = 2EC
And because DE is parallel to BC, then BD / DA = EC / AE
Then
BD / 12 = EC / 2EC
BD / 12 = 1/2
BD = 12 * (1/2) = 6
+56 
