KöMaL 2018 K596

Let $P$, $Q$ and $R$ denote points on sides $AB$, $BC$ and $CA$ of a triangle $ABC$, respectively, such that $AP = AR$, $BP = BQ$ and $CQ = CR$ should hold. How many different sets of such points $P$, $Q$, $R$ may exist for a given triangle $ABC$?