Q:

Does the angle bisector always pass through the midpoint of any line segment between the two sides of the angle? 

Consider this image: 

will the angle bisector of angle AOB always pass through the midpoint of AB, regardless of the lengths of AO and BO? 

A:

To be specific, if the angle bisector hits $AB$ in $P$ then we have the wonderful theorem
$$ PA:BP = OA:OB.$$
Thus $P$ is is the middle of $A$ and $B$ if and only if $OA=OB$ (i.e. the triangle is isosceles).