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 |
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Keywords
- Absolute geometry
- Hilbert geometry
- The Möbius-Pompeiu inequality
ASJC Scopus subject areas
- Mathematics(all)
Cite this
On a paper of dan barbilian. / Pambuccian, Victor.
In: Note di Matematica, Vol. 29, No. 2, 2009, p. 29-31.Research output: Contribution to journal › Article
}
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 -