On a paper of Dan Barbilian


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^{\circ}.
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].

DOI Code: 10.1285/i15900932v29n2p29

Hilbert geometry; the Mòbius-Pompeiu inequality; absolute geometry

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