Two circles and only a straightedge

TL;DR

This paper addresses Hilbert's question by proving the general impossibility of constructing the centers of two arbitrary circles with only a straightedge, while identifying specific circle pairs where such constructions are feasible.

Contribution

It establishes the general impossibility result and provides infinitely many examples of circle pairs allowing straightedge constructions of their centers.

Findings

General impossibility of center construction for arbitrary circles

Existence of infinitely many circle pairs with constructible centers

Clarification of geometric constraints in classical construction problems

Abstract

We answer a question of David Hilbert: given two circles it is not possible in general to construct their centers using only a straightedge. On the other hand, we give infinitely many families of pairs of circles for which such construction is possible.