in the unit circle confine two opposite arcs whose average is t" is equivalent to Cos[1/2 (-a + b - c + d)] == (Cos[a - b] - Cos[b - c] - Cos[a - d] + Cos[c - d])/ (2 Sqrt[(1 - Cos[a - c]) (1 - Cos[b - d])]) where a,b,c,d are angles on the circle. Is there a 3D analog of this theorem? Something like a cone with apex within the unit sphere cuts out two patches whose areas average to some function of the cone's solid angle? A couple of pictures look unpromising. --rwg Mma weirdness: In[1264]:= FindInstance[-1 == Sign[Sin[a - b] + Sin[b - c] + Sin[c - a]], {a, b, c}, Reals] Out[1264]= {{a -> 8/5, b -> 2, c -> 223/10}} OK, but In[1265]:= FindInstance[-1 == Sign[Sin[a - b] + Sin[b - c] + Sin[c - d] + Sin[d - a]], {a, b, c, d}, Reals] takes forever(?) finding nothing(?), even though In[726]:= Table[Sign[Sin[a - b] + Sin[b - c] + Sin[c - d] + Sin[d - a]] /. {a -> RandomReal[], b -> RandomReal[], c -> RandomReal[], d -> RandomReal[]}, {foo, 9}] Out[726]= {1, -1, 1, -1, -1, 1, 1, 1, -1} Presumably 50%.