On pentagons incribed in closed convex curve

The following two theorems are proved in the paper.
1. Suppose that the sum of any two neighbouring angles of a pentagon $A_1\dots A_5$ exceeds $\pi$. Let $A_0$ be any point on the boundary $\partial K$ of a convex set $K\subset\mathbb R^2$. Then there is an affine image of that pentagon such that this image is inscribe in $K$ and $A_0$ is the image of $A_1$.
2. The above theorem does not admit generalization to all pentagons inscribed in an ellipse.
3. Let $A_1,\dots,A_5$ be points of some ellipse, let $K\subset\mathbb R^2$ be a convex set with $C^4$-smooth boundary $\partial K$ of positive curvature, and let $A_0\in\partial K$ be a distinguished point of the boundary. Then there is an affine image of the pentagon $A_1\dots A_5$ such that this image is inscribed in $K$ and $A_0$ is the image of $A_1$.