Why two triangles are similar if they pass through median of bigger triangle?

Question

Qn: Let the cevians AD, BE, CF be concurrent in a triangle ABC. Prove that F E is parallel to BC if and only if AD is the bisector of BC.

Attempt1: Since midpoint gives similar base, two halves have similar area.
using this fact and some other facts of triangle, how can I prove that lines FE
parallel to BC iff AD is median?

Answer 1

Hint: If $FE||BC$ then $$\frac{FA}{FB}=\frac{EA}{EC}$$ and using ceva theorem that $$\frac{AF}{FB}\cdot \frac{BD}{DC}\cdot\frac{EC}{EA}=1$$