How many different ways can a circle intersect a triangle N ways?

Consider a circle intersecting a triangle. The circle and triangle can have between 0-6 total intersection points. Is there a mathematical formula for the number of possible ways they can intersect N number of times? This may require knowledge of how many of those N intersections are vertex intersections versus intersections with the side of the triangle.

For example, they can intersect at zero points in two ways, either the circle is entirely inside of the triangle, or entirely outside.

They can intersect at 1 point in 3 ways, either the circle touches a triangles vertex, the circle is outside of the triangle and intersects at a tangent point, or the circle is inside of the triangle and touches as a tangent point. The latter two cases are equally as important to me.

Following this logic, I believe there are 2 ways for N=5 intersection points, and there is 1 way for N=6 intersection points.

I ask this question because I am attempting to draw all possible ways as part of a larger personal project, and am not sure if I have found all cases for N=2-4.

Ways a triangle can intersect a circle

Thank you

• for each case, you can prove if you found them all or not. "they can intersect at zero points in two ways, either the circle is entirely inside of the triangle, or entirely outside" : ok how do you prove that ? :-) – reuns Apr 16 '16 at 0:42
• Hi, can you be a bit more specific about how to prove I have found them all for each case? I can see how you could show potential solutions for each case, but I'm not sure how it can be proven that every possible case has been found. – kreeser1 Apr 16 '16 at 2:29