Triangle Bisector

Question

$\overline{BE}$ and $\overline{CF}$ are angle bisectors of $\triangle ABC$ that meet at $I$, and we have $CE = 4$, $AE = 6$, and $AB = 8$. Find $BF$.

Answer 1

Expanding the hint given in the comments, the bisector theorem gives $$\frac{AE}{EC}=\frac{AB}{BC},$$ hence in our case $BC=\frac{16}{3}$. By the bisector theorem again, $$ BF = \frac{BC}{BC+CA}\cdot AB = \frac{\frac{16}{3}}{\frac{16}{3}+10}\cdot 8 =\color{red}{\frac{64}{23}}.$$