[Math] Determine the centre of a circle

Question

Given a circle $O$ on a paper, we do not know the centre point. Can we draw the centre only using a ruler (by which we can only draw straight lines)?

One fact I know: we can draw the tangent line at a point on the circle, by Pascal theorem.( or Noether's Max Fundamental theorem )

Thanks.

Answer 1

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.

Cut-the-knot has a simple proof that constructing the center of a circle using just a straightedge is impossible.