Build a Triangle II

Given: two line segments (blue, green) whose lengths are fixed and known. One of these line segments (blue) goes to the midpoint of the other (green). Given also an angle of fixed size (purple). Is it {always, sometimes, never} possible to build a triangle such that one of the line segments forms a side and the other a median to that side drawn from the purple angle? What questions could / would you put to your student based on this applet?