On quadrangles inscribed in a closed curve and the vertices of the curve

Let $ADCDE$ be a pentagon inscribed in a circle. It is proved that if $\gamma$ is a $C^4$-generic smooth convex planar oval with 4 vertices (stationary points of curvature), then there are 2 similarities $\varphi$ such that the quadrangle $\varphi(ABCD)$ is inscribed in $\gamma$ and the point $\psi(E)$ lies inside $\gamma$, as well as 2 similarities $\psi$ such that the quadrangle $\psi(ABCD)$ is inscribed in $\gamma$ and $\psi(E)$ lies outside $\gamma$. It is also proved that any circle $\gamma\hookrightarrow\mathbb R^n$ smoothly embedded in the space $\mathbb R^n$ of odd dimension contains the vertices of an equilateral $(n+1)$-link polygonal line lying in a hyperplane of $\mathbb R^n$.