Steinhaus' Problem on Partition of a Hyperbolic Triangle

Yıl 2018, Cilt: 9, 98 - 102, 28.12.2018

Oğuzhan Demirel

Öz

Hugo Steinhaus ([1],[2]) has asked whether inside each acute triangle there is a point from which perpendiculars to the sides divide the triangle into three parts of equal areas. In this paper, we present a solution of this problem in the Poincaré disc model of hyperbolic geometry.

Anahtar Kelimeler

Steinhaus' problem, hyperbolic triangle, hyperbolic quadrilateral, Gauss-Bonnet theorem

Kaynakça

• Ahlfors, L., Complex analysis: An introduction to the theory of analytic functions of one complex variable, Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978.
• Beardon, A.F., The Geometry of Discrete Groups, vol. 91 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1983.
• Ehrmann, J.P., Constructive solution of a generalization of Steinhaus’ problem on partition of a triangle, Forum Geom., 7(2007), 187–190.
• Ehrmann, J.P., On an affine variant of a Steinhaus’ problem, Forum Geom., 8(2008), 1–5.
• Steinhaus, H., Problem No. 132 (in Polish), Roczniki Polskiego Towarzystwa Matematycznego (Annales Societatis Mathematicae Polonae), Seria II, Wiadomosci Matematyczne, 9(1966), 99.
• Steinhaus, H., Problem No. 779 (in Polish), Matematyka, 19(1966), 92. Tyszka, A., Steinhaus’ problem cannot be solved with ruler and compass alone (in Polish), Matematyka, 49(1996), 238–240.
• Tyszka, A., Steinhaus’ problem on partition of a triangle, Forum Geom., 7(2007), 181–185.