Example 3 If 20 points divide a circle into 20 equal parts, then the number of regular polygons that can be formed with vertices chosen only from these 20 points is ( ).
(A) 4
(B) 8
(C) 12
(D) 24
(1996, National Junior High School Mathematics League)
Explanation: If you draw the figure directly, it is hard to get the correct answer. If you consider the characteristics of a regular polygon, list a simple equation for the distribution characteristics of points other than the vertices of the regular polygon, and then conduct a case analysis, it is easy to get the answer.

Let a regular $k$-sided polygon satisfy the condition. Then, the $20-k$ points other than the $k$ vertices are uniformly distributed on the minor arcs opposite to the sides of the regular $k$-sided polygon. Therefore, $\frac{20-k}{k}=\frac{20}{k}-1$ is an integer. Hence, $k \mid 20$.
Since $k \geqslant 3$, therefore, $k=4$ or 5 or 10 or 20. Thus, the number of regular polygons sought is
$$
\frac{20}{4}+\frac{20}{5}+\frac{20}{10}+\frac{20}{20}=12 \text {. }
$$

Therefore, the answer is (C).