Pentagons inscribed in a closed convex curve
The following two theorems are proved in the paper: 1. Suppose that the sum of any two neighbouring angles of a convex pentagon \(A_1,\dots,A_5\) exceeds \(\pi.\) Let \(A_0\) be a fixed point on the boundary \(\partial K\) of a convex set \(K\subset\mathbb R^2.\) Then there exists an affine image of this pentagon, inscribed in \(K,\) such that \(A_0\) is the image of \(A_1.\) This theorem does not admit generalization to all pentagons inscribed in an ellipse but the following theorem holds: 2. Let \(A_1,\dots,A_5\) be the 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 fixed point of the boundary. Then there exists 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 \(A_1.\)
scientific article; zbMATH DE number 1193408