It seems arguable to me whether geometrically a circle with an infinite radius is a line. For Graphics[Circle[{0, 0}, Infinity]] or even Graphics[Circle[{x, y}, Infinity]], conceived as the limit of Graphics[Circle[{x, y}, r]] as r -> Infinity, might be interpreted to represent the line at infinity in the projective plane. In the Euclidean plane, the limit of the circle has no meaning.

It seems to me that one obtains a line as the limit of a circle in the algebraic setting f[x, y, a] == 0, where f[...] == 0 gives the equation of a circle depending on a parameter a, which might be the radius or have some other relationship to the circle such as the reciprocal of the radius.

For as instance the radius 1/a in the following goes to infinity (as a -> 0), the circle becomes a line, which might be the sort of demonstration you're after:

Manipulate[
 ContourPlot[
  a x^2 - 2 x + a y^2 == 0, {x, -0.2, 4.}, {y, -2.1, 2.1}],
 {{a, 1.}, 0., 2.}
 ]