Problem

Source: 2020 USOJMO Problem 2

Tags: USA(J)MO, geometry



Let $\omega$ be the incircle of a fixed equilateral triangle $ABC$. Let $\ell$ be a variable line that is tangent to $\omega$ and meets the interior of segments $BC$ and $CA$ at points $P$ and $Q$, respectively. A point $R$ is chosen such that $PR = PA$ and $QR = QB$. Find all possible locations of the point $R$, over all choices of $\ell$. Proposed by Titu Andreescu and Waldemar Pompe