You don't have a circle until you have a circumference and a radius. Also, you have to decide if your circle is on a plane, meaning that the subset of interior points are all on the same plane. Then the circle is the set of points in a plane that form a closed curve so that every point of the curve is the same distance from the center of the circle which is a single point.

If your circle complies with those postulates it does not have an infinte radius or an infinte circumference.

Now let's forget those constraints and try to imagine circle with an infinite circumference. If you are at the center of the circle and want to travel to the circumference you will never be able to get there because it is infinitely far away.

If you were to start out at the circumference and wanted to travel to the center, you would never be able to get there either for the same reason.

You imaginary circle cannot exist in

incidence geometry.

Both of those statements about this imaginary circle are true because if you are at the center point, in order for the circumference to be an infinitely long closed curve it would have to have an infinitely long radius, and we have already claimed that no line with a start point and an end point, like a radius of the circle, can be infinite. Any straight line between two points in a plane is of finite length.

You can have any number of infinite lines all originating from one point, even an infinite number of such lines, and any two of those lines will have a finite angle between them. If you never attempt to measure the distance between any two of them, you can say that the distance between each of them increases toward infinity. However, even if those lines are all infinitely long, the distance between any two of them will never reach infinity. You cannot have an infinite distance between any two of them, because that distance implies a line that connects a point on one of the lines to a point on another of those lines in order to make the measurement. You see, your lines can extend infinitely, but by establishing a distance between them you have to also establish a point on each of them, and those two points are going to be a finite distance from the center point, and the line between them will always be finite. The "infinity" in that case is simply that the distance between each line increases infinitely.

This is correct, and though you can define a line that is infinitely long, and start that line at a point, once you establish a point somewhere else along the infinite line, the distance between the points can never be infinite.

No, now you are not thinking this through. You can have two infinite lines with the same start point that have an angle between them. You just cannot have an infinite distance between the far end of those lines because infinite lines do not have two ends.