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!

A circle of radius $r$ whose center is at $(r,0)$ has the parametric form
$$
\begin{array}{}x=r(1-\cos(\theta/r))&y=r\sin(\theta/r)\end{array}\tag{1}
$$
the limit of the curve in $(1)$ as $r\to\infty$ is
$$
\begin{array}{}x=0&y=\theta\end{array}\tag{1}
$$
which is the vertical line in your image.
Addendum:
In Inversive Geometry, circles and lines are considered the same. The inverse of a circle which passes through the center of the inversion is a line which doesn't pass through the center and vice-versa. The inverse of a line which passes through the origin is the line itself.
In the following image, the red and green circles are inverses with respect to the grey circle. Notice that when the red circle passes through the center of the inversion, the green circle becomes a line.

