### Abstract

We point out that the axiomatic analysis of the statement The segments joining a point with the vertices of an equilateral triangle satisfy the (non-strict) triangle inequalities in Barbilian's [1] misses the case in which the sum of the angles in a triangle is greater than 180°. We situate the statement correctly inside absolute geometry. We also point out that [1] contains the first proof that a Hilbert geometry with symmetric perpendicularity must be hyperbolic geometry, a proof commonly attributed to P. J. Kelly and L. J. Paige [5].

Original language | English (US) |
---|---|

Pages (from-to) | 29-31 |

Number of pages | 3 |

Journal | Note di Matematica |

Volume | 29 |

Issue number | 2 |

DOIs | |

State | Published - 2009 |

### Fingerprint

### Keywords

- Absolute geometry
- Hilbert geometry
- The Möbius-Pompeiu inequality

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Note di Matematica*,

*29*(2), 29-31. https://doi.org/10.1285/i15900932v29n2p29

**On a paper of dan barbilian.** / Pambuccian, Victor.

Research output: Contribution to journal › Article

*Note di Matematica*, vol. 29, no. 2, pp. 29-31. https://doi.org/10.1285/i15900932v29n2p29

}

TY - JOUR

T1 - On a paper of dan barbilian

AU - Pambuccian, Victor

PY - 2009

Y1 - 2009

N2 - We point out that the axiomatic analysis of the statement The segments joining a point with the vertices of an equilateral triangle satisfy the (non-strict) triangle inequalities in Barbilian's [1] misses the case in which the sum of the angles in a triangle is greater than 180°. We situate the statement correctly inside absolute geometry. We also point out that [1] contains the first proof that a Hilbert geometry with symmetric perpendicularity must be hyperbolic geometry, a proof commonly attributed to P. J. Kelly and L. J. Paige [5].

AB - We point out that the axiomatic analysis of the statement The segments joining a point with the vertices of an equilateral triangle satisfy the (non-strict) triangle inequalities in Barbilian's [1] misses the case in which the sum of the angles in a triangle is greater than 180°. We situate the statement correctly inside absolute geometry. We also point out that [1] contains the first proof that a Hilbert geometry with symmetric perpendicularity must be hyperbolic geometry, a proof commonly attributed to P. J. Kelly and L. J. Paige [5].

KW - Absolute geometry

KW - Hilbert geometry

KW - The Möbius-Pompeiu inequality

UR - http://www.scopus.com/inward/record.url?scp=84887203537&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887203537&partnerID=8YFLogxK

U2 - 10.1285/i15900932v29n2p29

DO - 10.1285/i15900932v29n2p29

M3 - Article

AN - SCOPUS:84887203537

VL - 29

SP - 29

EP - 31

JO - Note di Matematica

JF - Note di Matematica

SN - 1123-2536

IS - 2

ER -