An obvious necessary condition is that |a|, |b| and |c| can be sides of a triangle. The two-dimensional case can be analyzed further geometrically. If the triangle inequalities are satisfied, there are vectors congruent to a and b which form a triangle with c. The condition you require is that these vectors can be obtained by rotating a and b in opposite directions, a process which does not change the bisector of the angle between a and b.
Michael Renardy
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