Starting with an arbitrary complex number z, we will introduce a construction of a polygon P^{(1)}_z derived from a given polygon P. The inductively constructed sequence (P^{(k)}_z), associated to z and P, is studied, and its geometric properties are investigated. The complex numbers z for which the sequence (T^{(k)}_z) associated to a triangle T is “regular” are characterized, and the same is done for the sequence (Q^{(k)}_z) associated to a quadrilateral Q. By suitable choices of z, also the well known Napoleon theorem and some of its generalisations can be detected from the above characterizations.
Polygons derived from polygons via iterated constructions
VITALE, GAETANO;VINCENZI, Giovanni
2016-01-01
Abstract
Starting with an arbitrary complex number z, we will introduce a construction of a polygon P^{(1)}_z derived from a given polygon P. The inductively constructed sequence (P^{(k)}_z), associated to z and P, is studied, and its geometric properties are investigated. The complex numbers z for which the sequence (T^{(k)}_z) associated to a triangle T is “regular” are characterized, and the same is done for the sequence (Q^{(k)}_z) associated to a quadrilateral Q. By suitable choices of z, also the well known Napoleon theorem and some of its generalisations can be detected from the above characterizations.File in questo prodotto:
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