Determine the centre of a circle

According to wikipedia the answer is no:

In geometry, the Poncelet–Steiner theorem on compass and straightedge construction states that whatever can be constructed by straightedge and compass together can be constructed by straightedge alone, if given a single circle and its centre. This result is the best possible; if the centre of the circle is not given, it cannot be constructed by a straightedge alone.

Ok - it's too long to comment. I speak of possible additions to make it possible, as Gerry has indicated that it's not.

There is a big list of really small things that would make it possible. Having a ruler, rather than a straightedge would work. Or a straightedge with only a few marks (if allowed to make them ourselves beforehand - just 2 or 3, depending on whether the ruler is finite or not, to make right angles). Or the ability to make right angles. Alternatively, the ability to draw a line parallel to a given line (although this is cheap, as it's equivalent to finding midpoints of line segments). Or an inscribed parallelogram (really, we need two secants to the circle that are not parallel to eachother, and one line parallel to each to do this easily). If we don't mind building up a sweat, a circle and a parallelogram (anywhere) will do the trick too.

Or one secant and a parallel, and either one mark anywhere on the ruler. If we can make 1 mark anywhere on the ruler, then we don't even need the secant or its parallel (to distinguish - if the ruler comes with a mark on it, we need more. If we can place a mark on it, i.e. we can match one given length, we don't need anything more). Or another circle that intersects this circle in two places. Or a piece of string of fixed length (able to trace an arc of one particular curvature).

There are many more, I'm sure. But what I'm trying to get at is that this seems to be a good cusp - almost any additional information is enough information.