Actually I got it, the answer is 8.
Number the points clockwise, beginning with the upper left as 1. Number the center point 7.
We can create six equilateral triangles with side length one: 176, 172, 273, 657, 574, and 473.
We can also create two equilateral triangles with side length sqrt(3) : 135 and 246.
Thus, there are 8 such equilateral triangles.