Abstract
Two results, proved synthetically in plane Euclidean geometry by Chisini in 1924—stating that: (i) if MAB is an isosceles triangle, with \({MA\equiv MB}\), inscribed in a circle \({\mathcal{C}}\), P 1 and P 2 are two points on \({{\mathcal C}}\) such that {B, P i } separates {A, M} for \({i \in \{1,2}\}\), and {B, P 2} separates {M, P 1}, then \({AP_1+BP_1 < AP_2+BP_2}\), and (ii) of all triangles inscribed in a given circle the equilateral triangle has the greatest perimeter—are proved inside Hilbert’s absolute geometry.
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Chisini O.: La dimostrazione geometrica di un teorema di minimo. Periodico di matemtiche 5(4), 86–87 (1925)
Greenberg, M.J.: Euclidean and non-Euclidean geometries, 4th edn. W. H. Freeman, San Francisco (2008)
Hajja M.: Another morsel of Honsberger. Math. Mag. 83, 279–283 (2010)
Honsberger, R.: Mathematical gems III. Mathematical Association of America, Washington, DC (1985)
Honsberger, R.: Mathematical morsels. Mathematical Association of America, Washington, DC (1978)
Padoa A.: Una questione di minimo.. Periodico di matematiche 5(4), 80–85 (1925)
Padoa A.: Postilla ad una questione di minimo. Periodico di matematiche 6(4), 38–40 (1926)
Pambuccian V.: Zur Existenz gleichseitiger Dreiecke in H-Ebenen. J. Geom. 63, 147–153 (1998)
Schwabhäuser, W., Szmielew, W., Tarski, A.: Metamathematische Methoden in der Geometrie. Springer Verlag, Berlin (1983)
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Pambuccian, V. Absolute geometry proofs of two geometric inequalities of Chisini. J. Geom. 108, 265–270 (2017). https://doi.org/10.1007/s00022-016-0339-x
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DOI: https://doi.org/10.1007/s00022-016-0339-x