Generalizing the Petr-Douglas-Neumann theorem on \(n\)-gons. (English) Zbl 1053.51010

Gray, Stephen B.

Am. Math. Mon. 110, No. 3, 210-227 (2003).

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.

Reviewer: Grosio Stanilov (Sofia)

MSC:

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

Keywords:

Petr’s theorem; Napoleon-Douglas-Neumann theorem