The next certificate cannot have 5 or 6 intersections.

First of all, when this circle is inscribed with triangles of 3, 4 and 5, there are three intersections.

Another line and circle have at most two intersections, so there are at most six intersections.

But this circle can't have two intersections on each side, so there are six possibilities.

If there are five, then the numbers intersecting with the three sides are 2, 2, 1 respectively.

That is, tangent to one side. If we discuss it in three situations, it is easy to get this situation.

In summary, there are at most 4 intersections.