A circle with infinite radius is a line I am curious about the following diagram: 

The image implies a circle of infinite radius is a line. Intuitively, I understand this, but I was wondering whether this problem could be stated and proven formally? Under what definition of 'circle' and 'line' does this hold?
Thanks!

There is no such thing as a circle of infinite radius. One might find it useful to use the phrase "circle of infinite radius" as shorthand for some limiting case of a family of circles of increasing radius, and (as the other answers show) that limit might give you a straight line. 
