### Abstract

An n-gon is called Napoleon if the centers of the regular n- gons erected outwardly on its sides are vertices of a regular n-gon. In this paper we obtain a new geometric characterization of Napoleon n-gons and give a new proof of the well-known theorem of Barlotti-Greber ([1], [4]) that an n-gon is Napoleon if and only if it is affine-regular. Moreover, we generalize this theorem by obtaining an analytic characterization of the n-gons leading to a regular n-gon after iterating the above construction k times.

Original language | English |
---|---|

Pages (from-to) | 24-29 |

Number of pages | 6 |

Journal | American Mathematical Monthly |

Volume | 122 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*American Mathematical Monthly*,

*122*(1), 24-29. https://doi.org/10.4169/amer.math.monthly.122.01.24

**Napoleon polygons.** / Andreescu, Titu; Gueorguiev, Vladimir Simeonov; Mushkarov, Oleg.

Research output: Contribution to journal › Article

*American Mathematical Monthly*, vol. 122, no. 1, pp. 24-29. https://doi.org/10.4169/amer.math.monthly.122.01.24

}

TY - JOUR

T1 - Napoleon polygons

AU - Andreescu, Titu

AU - Gueorguiev, Vladimir Simeonov

AU - Mushkarov, Oleg

PY - 2015

Y1 - 2015

N2 - An n-gon is called Napoleon if the centers of the regular n- gons erected outwardly on its sides are vertices of a regular n-gon. In this paper we obtain a new geometric characterization of Napoleon n-gons and give a new proof of the well-known theorem of Barlotti-Greber ([1], [4]) that an n-gon is Napoleon if and only if it is affine-regular. Moreover, we generalize this theorem by obtaining an analytic characterization of the n-gons leading to a regular n-gon after iterating the above construction k times.

AB - An n-gon is called Napoleon if the centers of the regular n- gons erected outwardly on its sides are vertices of a regular n-gon. In this paper we obtain a new geometric characterization of Napoleon n-gons and give a new proof of the well-known theorem of Barlotti-Greber ([1], [4]) that an n-gon is Napoleon if and only if it is affine-regular. Moreover, we generalize this theorem by obtaining an analytic characterization of the n-gons leading to a regular n-gon after iterating the above construction k times.

UR - http://www.scopus.com/inward/record.url?scp=84995554773&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84995554773&partnerID=8YFLogxK

U2 - 10.4169/amer.math.monthly.122.01.24

DO - 10.4169/amer.math.monthly.122.01.24

M3 - Article

VL - 122

SP - 24

EP - 29

JO - American Mathematical Monthly

JF - American Mathematical Monthly

SN - 0002-9890

IS - 1

ER -