[math-fun] Is this locus smooth everywhere?

Let P = (-1,0) and Q = (1,0) in the plane. Let L be the locus of all points X such that the angle PXQ is 120 degrees. Include in L the points P and Q as well. QUESTION: Does L have a well-defined tangent line at P and Q ??? (And if so, is it C^oo or C^w (real analytic) at P and Q as well?) --Dan