Shandong's questions are very difficult, and I always feel that Shandong's questions are very difficult. This question examines many knowledge points such as vertical diameter theorem, circle angle theorem, pythagorean theorem, the properties of equilateral triangle, the judgment and properties of rectangle and so on. It is quite difficult to examine creative thinking. The key is to construct an auxiliary circle. Look at the answer/exercise/math /798720. I hope you can adopt it.

The coordinates of point A and point B are (1, 0) and (5,0) respectively, and point P is the moving point in the rectangular coordinate system.

(1) There are countless points P such that ∠ APB = 30;

(2) If point P is on the Y axis and ∠ APB = 30, find the coordinates of point P;

(3) Does ∠APB have a maximum when point P moves on the Y axis? If so, find the coordinates of point P, and explain why ∠APB is the largest at this time.