Petr-Douglas-Neumann theorem

If isosceles triangles with apex angles 2k π/n are erected on the sides of an arbitrary n-gon A_0, and if this process is repeated with the n-gon A_1 formed by the free apices of the triangles, but with a different value of k, and so on until all values 1<=k<=n - 2 have been used in arbitrary order, then a regular n-gon A_(n - 2) is formed whose centroid coincides with the centroid of A_0. Napoleon's theorem and van Aubel's theorem are special cases of the Petr-Neumann-Douglas theorem.

Douglas-Neumann theorem | isosceles triangle | Napoleon's theorem | van Aubel's theorem