Well, in my opinion this question is as simple as it can get. I am not aware of a rule that describes this method. The only TRAP for which even I fell is the option 4.

So, the number of intersection between a triangle and a circle can be the following numbers:
1: Imagine a triangle tip touching a circle.
2. A triangle with one side outside the circle. (2 sides intersecting the circle).
3: Circumcircle or Incircle ( https://images.google.co.in/images?q=Circumcircle )
4: Imagine a circumcircle scenario with only one vertex (out of 3) outside the circle. The rest two are on the circle.
5: Imagine a circumcircle scenario with two vertices (out of 3) outside the circle. Only one is on the circle.
6: All the vertices of the triangle are outside the circle. Surprisingly, I found an image on google. I dint want to draw one.
( https://garethrees.org/2009/02/19/circle ... /cases.png ) - Check Image 5 for this case 6.