Matchstick Problem - 2

It's possible to make 8 equilateral triangles. I will show that we can't make >8.

Draw any old line and call it the x-axis. We can group the matchsticks into classes based on which of the intervals $[0,\frac{2\pi}{3})$ , $[\frac{2\pi}{3},\frac{4\pi}{3})$ , $[\frac{4\pi}{3},2\pi)$ the angle they make with the x axis falls into . Let the number of matchstick in each class be $a_{1}$ , $a_{2}$ , $a_{3}$ . Any equilateral triangle must have one side from each class. So the number of equilateral triangles is at most $a_{1}a_{2}a_{3} \leq (\frac{a_{1}+a_{2}+a_{3}}{3})^{3} = (\frac{6}{3})^{3} =8$ (by AM-GM).