I have to agree with Templar on this. (Though I disagree with his comment about "then they much must meet at infinity" since meeting at infinity is essentially a concession that they are indeed parallel for all practical purposes, which they certainly are not.)
Your example is simply a conceptual illusion hyparzero... playing on the fact that infinity 1) may have different measures under different circumstances, 2) is really a concept and not a number, and can never be reach in practice - hence it's actually not practical to 'visualise' the situation which you proposed.
You used a circle simply to disguise what's really at issue here - any other closed geometric shape would have also done the trick: square, rhombus, etc, you name it.
The point of 'infinity' is that it doesn't have a boundary, yet you're giving us examples of constrained geometric figures with boundaries and asking us to imagine stretching it to infinity - imagine that: a figure stretching everywhere to infinity and which also has a boundary.
It's just as well graphs and diagrams are not accepted as 'proofs' by professional mathematicians, as they tend to be the substance of fallible intuition. When putting pen to paper, however, you won't be able to prove your example mathematically. Since any shape stretched to infinity flatly is a self-contained Euclidean plane - and there is only one such parallel line that exists through a point outside a particular line.
But of course, you already knew that.
Let us put your proposition then like this:
Let there exist a circle initially of finite radius, R, embedded in a flat plane. Then, as you said, suppose we perform a continuous (i.e. not discrete) transformation on the circle so as to increased its radius of curvature, and thereby its circular boundary, whilst maintaining the circular shape (which is defined by its everywhere-constant curvature along the boundary), for infinite time.
We are now in a situation of three dilemmas, and indeed three alternatives of interpretation:
1) We may assume that the circle is only defined by its boundary and nothing else - i.e. as a circular path, we are not interested in the number of [mathematically dis-connected] regions it divides the flat plane into, but only the path itself.
Then we end up with this: there exists no well-defined lines 'inside' or 'outside' the circle, because there is no well defined regions of interest.
Hence, the boundary (which defines the circle itself) is the only object of existence - and any boundary is composed of one single line, parallel and coincident to itself.
2) We think of "the circle" as a composition (i.e. a mathematical union) of i) the boundary, and ii) the interior of the boundary, which is an inner region.
Thus, when we say we are continually transforming "the circle" by increasing its radius of curvature, we are saying that BOTH the boundary and the inner region are being transformed under the same transformation (e.g. to understand this, consider a circle as an infinite union of circles of radius 0 up to R pivoted at the same centre).
Then we end up with this: as the circle starts 'expanding' from r = R to r = infinity continually, so does it's inner region. Therefore, if it is conceivable to 'visualise' or 'draw', as you ask us to, lines of infinite length at the moment in time when r = infinity, then it is conceivable that any such infinite line must correspond to a initial line interval in the inner region of the initial circle of radius R which has had the same transformation performed on it for the same amount of time:
Find me any two of the blue chords in your circle that does not both intersect with the black chord at any time after you begin an expansion of that circle! (the idea, of course, is that you can't!)
3) Suppose now, that your transformation isn't continuous and instead discrete.
Then we end up with this: you jumped from something finite to something infinite in one fell-swoop. This means the intial circle of radius R is disconnected in time to the circle of infinite radius.
Then i) the circle of infinite radius is simply an infinite flat plane, and ii) in order that you may 'draw' infinite parallel lines in such a flat plane you need to relate it to the initial circle of radius R.
But the initial circle is a separate discrete entity in time with a finite boundary and any chords in such a circle has finite length - i.e. the chords are line intervals, not lines. They do not even satisfy the basic criterion in the definition of "parallel-ness".
Hence, you cannot relate an infinite plane to a finite region in such a way and thus cannot reach the conclusion that it is possible for an infinite number of paralell lines to exist through a single point on an infinite plane.
Ergo, all three interpretations exclude the possibility for the existence of an infinite number of such parallel lines through any single point on an infinite Euclidean plane.
Conclusion: your 'visualisation' is the work of illusion.
P.S. Here's an illusory 'proof' that ALL lines are coincident at every point on the lines no matter how one draws them: (inspired by the 'logic' of your argument)
Using your circle again, let it expand 2-dimensionally as you said before with the blue and black chords draw in them. 'Imagine' the observer moving away from the expanding circle perpendicular to its plane at a rate faster than it is expanding.
Then at infinite time, the region is infinite in all directions and yet seen as a point to the observer moving quick away from it at a large distance away - hence, we conclude that all lines of infinite length on an infinite plane are point-like and so are coincident at every point on the lines.
Does that make sense?
If not, then so doesn't your argument.
