Problem #PRU-58300

Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Problems

Problem

Do there exist three points \(A\), \(B\) and \(C\) on the plane such that for any point \(X\) the length of at least one of the segments \(XA\), \(XB\) and \(XC\) is irrational?