zbMATH — the first resource for mathematics

Generalizing the Petr-Douglas-Neumann theorem on \(n\)-gons. (English) Zbl 1053.51010
The paper takes a different point of view, generalizing the PDN-theorem while using a proof method that is more elementary but necessarily simpler than existing proofs.
The following Theorem 1 is proved. Starting with an arbitrary plane polygon \(A_0\), using triangles from a model \(n\)-gon \(Z\), and constructing successive polygons \(A_k\) according to the general construction rule, the polygon \(A_{n-2}\) is similar to \(Z\) and \(A_{n-1}\) is a single point.

51M04 Elementary problems in Euclidean geometries
51M20 Polyhedra and polytopes; regular figures, division of spaces
52B11 \(n\)-dimensional polytopes
Full Text: DOI