jmc

Each of the $\left(\genfrac{}{}{0ex}{}{9}{2}\right)=36$ pairs of vertices determines two equilateral triangles, for a total of 72 triangles. However, the three triangles $A_1{A}_4{A}_7$, $A_2{A}_5{A}_8$, and $A_3{A}_6{A}_9$ are each counted 3 times, resulting in an overcount of 6. Thus, there are $\boxed{66}$ distinct equilateral triangles.