The Center of a Circle — Learn the “How-to Find”
Using a compass, you can draw a circle at any place, with any radius.
Now let’s reverse the problem. Given a circle, do you know how to find its center? (Of course, once the circle is found, there shall be no problem at all to tell its diameter, or radius.) You only see the circle itself, there is no explicit indication on where the center is.
With two set of restrictions on what kind of tools you can use, there are actually two questions. In general, you can find the center using any convenient method, including copy-and-paste the circle onto a paper, and then fold it. In particular (from classical Euclidean geometry), where it’s required to do so with a ruler (with which you are allowed to draw lines and line segments only) and a compass (with which you are allowed to draw circles only).
See the following article on how to do it in general.
If you attempt to solve this problem with a ruler and a compass, then you are required to know how to make a perpendicular bisector. This will be discussed in another posting.