Two circles and only a straightedge

The study of straightedge-and-compass geometric constructions goes back to ancient geometry. Many centuries later, the mathematicians became interested in straightedge-only constructions. Recall that a straightedge (a.k.a. a ruler without marks) is a device that can be used to connect any two points by a line but not to measure distances or draw parallel lines. Surprisingly, many things can be achieved with only a straightedge: for example, one can construct tangent lines to a given circle.

A famous German mathematician David Hilbert proved that given a circle, it is not possible to construct its center using only a straightedge. Certain vagueness in his proof led to an error that laid undiscovered for more than 100 years...