Since the obvious algorithm didn't work I decided to brute force through the first 20 primes except 2.
The result was(clockwise, starting at the bottom):
127 with 3, 17, 23, 7, 11, 19, 13, 5, 29
That are the first 9 primes after 2, so I guess I should have reached that without my computer.

I was curious and tested a bit further and the triangle seems to work for any sequence of 9 consecutive primes (that I tested). Is that a global fact or is it just valid for the first few cases? If yes, why?