Side of triangle problem

Question

In triangle $ABC$, $AB=BC=12$. Side $AC$ extended through $C$ a length equal to itself to a point $D$. Point $E$ is on $AB$; $DE$ intersects $BC$ at $F$ and $BF$ equal to 8. Find $AE$ without using the centroid.

Answer 1

Unless you directly see similar triangles, vectors is always the best approach at these kinds of things. An answer, unlike geometrical approach, is always guaranteed.

Assume $B$ at origin. Vectors $\vec a,\vec c$ are $\vec {BA},\vec{BC}$

All are vectors below :

$CA=\vec c-\vec a$

$BD=\vec c+\vec c-\vec a$

$BF=\frac{8}{12}\vec c $

$DF=BD-BF$

Assume $\vec{BD}+x\vec{DF}=y\vec {BA}$

Where $x\vec{DF}=\vec{DE}$ and $y\vec{BA}=\vec{BE}$ Compare magnitudes of $\vec a$ and $\vec c$ on both sides as they are independent.

Answer 2

Draw CP // AB.

$\triangle DPC$ is similar to $\triangle DEC$

Then, AE = (0.5)PC

$\triangle FPC$ is similar to $\triangle FEB$

Then, EB = (0.5)PC

Therefore, AE = EB = (0.5)AB = ... = 6