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

Question

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?

Answer 1

What you're asking is the following: in a given triangle, is the perpendicular bisector of some angle the median to the side in front of this angle?

The answer is clearly no, as the above condition characterises isosceles triangles...

Answer 2

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).