  We show that the $n$-division points of all rational hypocycloids are
constructible with an unmarked straightedge and compass for all integers $n$,
given a pre-drawn hypocycloid. We also consider the question of
constructibility of $n$-division points of hypocycloids without a pre-drawn
hypocycloid in the case of a tricuspoid, concluding that only the $1$, $2$,
$3$, and $6$-division points of a tricuspoid are constructible in this manner.
