$16 \cdot 6$ Find a point on the side of a rectangle such that the lines connecting this point to the endpoints of the opposite side divide the rectangle into three mutually similar triangles. The number of such points:
(A) Does not exist.
(B) There are definitely 2.
(C) There are definitely 4.
(D) The number cannot be determined.
(China Guangzhou, Wuhan, Fuzhou, and other five cities junior high school mathematics league, 1987)
[Solution] The lines connecting such a point with the two endpoints of the opposite side should form a $90^{\circ}$ angle, so such a point is the common point of the semicircle constructed inside the rectangle with one side of the rectangle as the diameter and the opposite side.
When $A B C D$ is a square, such a point does not exist, which negates $(B),(C)$; when $A B>2 A D$, there are four such points, which negates $(A)$. Clearly, ( $D$ ) is correct. Therefore, the answer is $(D)$.