Given that $$\triangle ABC$$ is an arbitrary triangle. The number of points $$P$$ in the interior of the triangle such that $$\angle APB = \angle BPC = \angle CPA$$ is [ ]

For an interior point the three angles add up to $$360^\circ$$, so each must be $$120^\circ$$. Such a point lies on the arc over $$AB$$ subtending $$120^\circ$$ and on the arc over $$BC$$ subtending $$120^\circ$$, so there is at most one of them, and it lies inside the triangle exactly when every angle of the triangle is less than $$120^\circ$$. If some angle, say $$\angle A \ge 120^\circ$$, then every interior point $$P$$ has $$\angle BPC > \angle A \ge 120^\circ$$, so no such point exists. Hence the number is 0 or 1 depending on the triangle.