9. 16 Color 5 vertices of a regular nonagon red. How many pairs of congruent triangles, whose vertices are all red, exist at least?
[Solution] There are $C_{5}^{3}=10$ triangles with 5 red points as vertices, which we call red triangles. Let the circumference of the circumcircle of the regular nonagon be 9, and use the length of the arc subtended by the chord to represent the chord length. In the regular nonagon, there are only 7 types of triangles that are not congruent to each other, with their 3 side lengths being: $(1,1,7),(1,2,6),(1,3,5),(1,4,4)$, $(2,2,5),(2,3,4),(3,3,3)$. Therefore, by the pigeonhole principle, there are at least 3 pairs of congruent red triangles. If there are no fewer than 3 red triangles that are pairwise congruent, then the number of congruent red triangle pairs is at least 4. Hence, we assume that different pairs are not congruent to each other.

Let two red triangles $\triangle A_{1} A_{2} A_{3}$ and $\triangle B_{1} B_{2} B_{3}$ be congruent. Since there are only 5 red points, these two red triangles must have at least 1 common vertex and at most 2 common vertices.
(1) Suppose the two congruent red triangles have exactly 1 common vertex. Since they are inscribed in the same circle, they must be symmetric about the diameter passing through the common vertex. Thus, the other 4 red points form the 4 vertices of an isosceles trapezoid. It is easy to see that in this case, there are at least 4 pairs of congruent red triangles.
(2) Suppose the two congruent red triangles have 1 common side. Since their vertices are all vertices of the regular nonagon, the line connecting the other two endpoints of these triangles is parallel to the common side. Thus, another isosceles trapezoid is formed, which contains two pairs of congruent red triangles. In addition to these two pairs, there is at least one more pair of congruent red triangles, and from this pair, another pair can be found. Clearly, this pair will not repeat with the previous two pairs, so there are at least 4 pairs of congruent red triangles.

In the figure on the right, the 10 red triangles formed by the 5 red points have exactly 4 pairs of congruent red triangles.

In summary, the minimum number of pairs of congruent triangles is 4.