What is the radius of the largest orthogonal circle (as a function of R)that can be constructed for a given circle of radius R?
Definition of Largest circle-the circle that spans the longest section of the circumference of given circle of radius R on it’s convex side
For any (true) circle orthogonal to a given circle, you can always find a larger one, with ever-more of the given circle's circumference on its "convex side". So, there is no "largest" (true) circle.
Now, larger and larger circles have smaller and smaller curvature; larger and larger orthogonal circles get closer and closer to the center of the given circle. Thus, this sequence of circles, as they say, "tend toward" a straight line through the given circle's center. It's sometimes philosophically helpful to consider a line to be a circle, just one of infinite radius. (A number of theorems become easier to state.) In that sense, the "largest circle" is that line through the center.