Example 5 As shown in Figure 9, given a square $A B C D$ divided into four smaller rectangles by two line segments $E F$ and $G H$ parallel to the sides, $P$ is the intersection of $E F$ and $G H$.
If the area of rectangle $P F C H$ is exactly twice the area of rectangle $A G P E$, determine the size of $\angle H A F$ and prove your conclusion.
(1998, Beijing Middle School Mathematics Competition (Junior

2))