If you have a line, and it goes on forever, and you choose a random point on that line, is that point the center of that line? And if you picked a new point, would that become the center of the line (since to either side of the point is infinity, and infinity is congruent to infinity)? Also if the universe has no middle and no end, am I, and everyone, at the center of the universe? (Of course the middle of the universe thing only works if you believe the universe has no middle and no end.)

As with so many questions in mathematics, the answer will depend on exactly how you define your terms. In this case, we will have to decide how to define the word "center". Now, you hint at a possible definition in your question, when you speak of the parts of the line on either side of a point as being congruent. Let's make this definition explicit. Suppose we define a center point of a line or a line segment to be a point with the property that the parts of the line or line segment on either side of that point are congruent. Then, for example, in a line segment of length 1 inch, the point that is 1/2 inch from each end will be the unique center point of the segment; the parts of the segment on either side of that point both have length 1/2, and are therefore congruent. But if we apply this definition to a line that extends infinitely far in both directions, then we find that every point is a center point, because, as you observe, the parts of the line on either side of any point extend infinitely far, and are therefore congruent to each other.

There is no contradiction or paradox here. With this definition of "center", an infinite line does not have a unique center point, it has many center points. I think what made this situation seem puzzling to you is that you were using a definition of "center" according to which a line has more than one center, but you also used the word "the", in the phrase "the center", which only makes sense if there is a unique center.

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