Gauss–Wantzel theorem states that
A regular n-gon is constructible with straightedge and compass if and only if $n = 2^kp_1p_2...p_t$, where $p_i$'s are distinct Fermat primes (A Fermat prime is a prime number of the form $2^{m}+1$).
A Poncelet polygon (or bicentric polygon) is a polygon which is simultaneously inscribed (all vertices lie on the same circle) and circumscribed (all sides are tangent to the same circle).
Such polygons play the main role in Poncelet's closure theorem (https://mathworld.wolfram.com/PonceletsPorism.html). Obviously, each regular polygon is a Poncelet polygon.
My question is: for which $n$ it is possible to construct a non-regular Poncelet n-gon with compass and straightedge?
I know an explicit construction for $n=2^k,3 \cdot 2^k, 5 \cdot 2^k$.
Remark. For each $n$ there is a relation between $R$ (radius of circumcircle), $r$ (radius of incircle) and $d$ (distance between incenter and circumcenter). For $n=3$ we have Euler formula, for $n=4$ we have Fuss formula, etc (see https://mathworld.wolfram.com/PonceletsPorism.html). So if we put $d=1$, the question is to find constructible numbers $R$ and $r$ satisfying the relation.