Example 5 In the plane of square $ABCD$, a line $l$ satisfies the following conditions: the distances from the four vertices of square $ABCD$ to line $l$ take only two values, one of which is three times the other. The number of such lines $l$ is ( ).
(A) 4 lines
(B) 8 lines
(C) 12 lines
(D) 16 lines
Solution: As shown in Figure 2.
(1) Four lines passing through the quartiles of one pair of opposite sides and parallel to the other pair of sides;
(2) Four lines passing through the points that externally divide one pair of opposite sides in the ratio $3: 1$ and parallel to the other pair of sides;
(3) Four lines passing through the midpoints of one pair of adjacent sides;
(4) Four lines passing through the quartiles of one pair of opposite sides and not parallel to the other pair of sides.
In summary, there are 16 lines $l$ that meet the requirements. Therefore, the answer is (D).