Problem

Source: Original RMM 2019 P4

Tags: geometry, Equilateral Triangle, circumcircle, contest problem



Let there be an equilateral triangle $ABC$ and a point $P$ in its plane such that $AP<BP<CP.$ Suppose that the lengths of segments $AP,BP$ and $CP$ uniquely determine the side of $ABC$. Prove that $P$ lies on the circumcircle of triangle $ABC.$