Prove an expression for angle bisector 
Show that the vector $\dfrac{A\,\vec B+B\,\vec A}{A+B}$ represents the bisector of the angle between $\vec A$ and $\vec B$.

I can prove that the numerator is the bisector of both vectors but I am unsure how to show that the expression given is as well. Does it matter that the expression is divided by a scalar? I would assume not, but I am not sure. Thanks.

You are right that the denominator is not that important. Here it serves to give a convex combination of the points $\vec A$ and $\vec B$, i.e., the bisector that is in the segment between these points.
