An affine-regular $n$-gon is a non-degenerate affine image of the regular $n$-gon. It seems to be a standard fact in combinatorial geometry that inside every convex compact set $K\subseteq \mathbb R^2$ with nonempty interior it is possible to inscribe at least one affine-regular hexagon so that its vertices belong to the boundary of $K$. This is stated, for example, here.
How does one prove this? I am not able to find an accessible reference that contains the proof. Is there "typically" a unique inscribed hexagon?