The answers to [this question](https://math.stackexchange.com/q/104887/23846)
imply that a line which bisects two sides of a triangle must be parallel to the third side. Why is this true? There must be a simple proof.

More generally: Let D and E be points on $\overline{AB}$ and $\overline{CB}$, respectively, such that $AD:DB=CE:EB$. Then, $\overleftrightarrow{DE} \parallel \overline{AC}$. (Thanks @Isaac)

![Bisectors of a Triangle](https://i.stack.imgur.com/lgl0p.png)

We could also look at the contrapositive. Suppose that $DF$ is not parallel to $AC$. Then 
the triangles $DBF$ and $ABC$ won't be similar.

![Contrapositive](https://i.stack.imgur.com/rAXLH.png)