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!

Algebraic description
A generic algebraic curve of degree $2$ is the set of points satisfying
$$ax^2 + by^2 + cxy + dx + ey + f = 0$$
You can define circles to be those curves with $a=b$ and $c=0$. You can compute the radius as
$$r=\sqrt{\frac{d^2 + e^2}{4a^2} - f}$$
For the case of $a=b=0$ your equation describes a line, and your radius becomes infinite due to the division by zero.
Note that the definition also includes circles with imaginary radius, which have no points in the real plane. You may exclude these by restricting the range of parameters.
Complex plane and cross ratios
Interpreting the points of your plane as complex numbers, you can define that four points lie on a common circle or line if their cross ratio is a real number. Using this definition, a line is just a circle, and the only reasonable value to assign to its radius is $\infty$.
Möbius Geometry
A Möbius transformation will map circles and lines in the complex plane (or complex line, depending on your use of the word) $\mathbb C$ to other circles and lines. So the above definition of a circle will suit these transformations as well. There is a scientific topic called Möbius Geometry which uses 4-dimenstional vectors to describe both lines and circles. In this setup again a line is a special case of circle, and performing any radius computation will lead to infinity.
You may extend Möbius geometry a bit further to obtain Lie geometry, where even points are special cases of circles, namely those with zero radius. Lie transformations may transform generic circles to lines or points and vice versa.
Constant curvature
You may define a circle as a line of constant curvature, and without endpoints. You may imagine curvature as the inverse of the radius, but you can define it in other ways as well. Obviously, for the case of zero curvature and therefore infinite radius, you will obtain a line.
Projective geometry
You can define a circle to be a conic incident with the two complex circle points
$$\begin{pmatrix}1\\i\\0\end{pmatrix}
\text{ and }
\begin{pmatrix}1\\-i\\0\end{pmatrix}$$
where the points are given in homogenous coordinates in $\mathbb{CP}^2$. There are conics with real coefficients which degenerate into a pair of lines. One of them is the line at infinity incident with the two points above, and the other may be a finite line. So unless your conic degenerates to a double line at infinity, you get a single line as the finite portion of a conic which by definition is a circle. You can compute the center of that circle using the points above, and the result will be an infinite point, indicating the infinite radius.
