First note that $n > 8$ because of this set of triangles: 123, 345, 567, 781, 247, 461, 683, 825 If there are 9 triangles, that makes 27 point-triangle relations, so by PHP some point must belong to 4 triangles, which would mean 8 edges coming from that point. There are only 7 other points to connect to, however, so one of those edges must be repeated, and therefore must be shared by two triangles. Therefore $n = 9$.