[math-fun] Reflecting a spherical triangle

Suppose ∆ is a spherical triangle on the 2-sphere S^2 with its angles A, B, C each equal to a rational multiple of π. Now suppose we reflect ∆ across each of its sides, and reflect each of the resulting triangles about each of its sides, etc., indefinitely until there remain no sides that haven't been reflected across. Let V be the set of all the vertices resulting from these reflections. Questions: * For which rational multiples {A, B, C} of π will the set V be finite? * If V is infinite, does that imply V is dense in S^2 ? —Dan