Change bisector from one triangle to another

[roni1]

LB is bisector of triangle ABC.
suppose, there is a new triangle that called AED.
alpha is a angle between LB to AD that is bisector of AED.
The triangles can be on any place in the plane.
What the value biggest angle ($$alpha$$) can be?

 

[skeeter]

LB is bisector of triangle ABC.
suppose, there is a new triangle that called AED.
alpha is a angle between LB to AD that is bisector of AED.
The triangles can be on any place in the plane.
What the value biggest angle ($$alpha$$) can be?

This description lacks clarity ... can you upload a sketch to better understand what you mean?