Example 9 In $\triangle A B C$, $\angle A>\angle B>\angle C$, $\angle A \neq 90^{\circ}$, draw a line that divides $\triangle A B C$ into two parts, and makes one of the parts similar to $\triangle A B C$. There are ( ) such non-parallel lines.
(A) 2
(B) 3
(C) 6
(D) 12
Solution: (1) There are 3 lines parallel to each of the two sides;
(2) There are 3 anti-parallel lines to the three sides (i.e., the four points of the quadrilateral formed are concyclic).
These 6 lines all meet the requirements of the problem. Therefore, the answer is (C).