I'm not sure we're saying the same thing. I'm saying that it is not true that it can only touch three (or four, or any less-than-infinite number of circles). I

I'm not sure we're saying the same thing. I'm saying that it is not true that it can only touch three (or four, or any less-than-infinite number of circles). I think you're saying that only three of them are necessary to draw the correct circle. The analogy would be that any three points on a plane describe both the vertices of exactly one triangle and exactly one circle circumscribing it (your statement) and that an infinite number of circumscribed triangles can be found for any nontrivial circle (my statement).