#### Answer

$BF = 6$
$DE = 6$

#### Work Step by Step

Because $\overline{BC}$ is bisected into $\overline{BF}$ and $\overline{FC}$, $BF$ should be half of $BC$, or $6$.
The diagram indicates that $\overline{DE}$ bisects both $\overline{AC}$ and $\overline{AB}$. According to the triangle midsegment theorem, if a line segment joins two sides of a triangle at their midpoints, then that line segment is parallel to the third side of that triangle and is half as long as that third side.
Knowing this information, we can deduce that $\overline{BC}$, which is the third side, is parallel to $\overline{DE}$, which is the line segment, and that $BC$ is two times the length of $DE$. This gives:
$BC = 2(DE)$
Plug in $12$ for $BC$:
$12 = 2(DE)$
Divide each side by $2$ to solve:
$DE = 6$