What is minimum radius of a circle tangent to three parallel lines?

Question

This cannot happen, of course, in eucludean geometry.

Answer 1

I would imagine that the answer must depend on what sort of geometry you are working with. Which geometry do you mean?

Seems like you would need negative curvature for this to happen, so probably you are tallking about the hyperbolic plane. For this, you may as well assume the disk model with a circle centered at the origin. I'm not going to work it out, but intuitively to me it seems like the minimum circle would be the one that is tangent to three "lines" which intersect pairwise at infinity (by infinity, I mean the edge of the disk).

Edit:

Okay I worked it out in the shower. In the standard hyperbolic plane I think the answer is tanh^{-1}(2-sqrt(3)). tanh^{-1} is the inverse hyperbolic tangent.

Answer 2

It can happen in Euclidean geometry.

Imagine three parallel lines in three-dimensional space. Since the lines are parallel, we can pick a plane P perpendicular to those lines, giving us 3 points in plane P. With those three points A, B, and C construct the triangle ABC. Finally, construct the inscribed circle of triangle ABC.

The radius of the circle you seek is the distance from the incenter to the point of tangency of the incircle and the triangle.

That would be more readily understood were I to draw it.