Journal for Geometry and Graphics 29 (2025), No. 1, 065--077
Copyright by the authors licensed under CC BY SA 4.0
Invariant Lines and a Projective Geometric Generalisation of a Geometry Problem
Peter Csiba
Department of Mathematics, J. Selye University, Komárno, Slovakia
csiba@ujs.sk
Dina Kamber-Hamzic
Department of Mathematics and Computer Science, University of Sarajevo, Bosnia-Herzegovina
dinakamber@pmf.unsa.ba
László Németh
Institute of Basic Sciences, University of Sopron, Hungary
nemeth.laszlo@uni-sopron.hu
Zenan abanac
Department of Mathematics and Computer Science, University of Sarajevo, Bosnia-Herzegovina
zsabanac@pmf.unsa.ba
Generalisations of mathematical problems often require innovative approaches and open up promising avenues for further research. In this article, we propose a generalisation of a geometry problem originally featured in the 1995 International Mathematical Olympiad contest. We employ projective geometric techniques to rigorously prove our generalisation, demonstrating the value of applying advanced mathematical tools to extend the boundaries of traditional problems. Even though this is a geometry problem originally intended for students, it opens up many interesting ideas for generalisation and the inclusion of more advanced tools to prove these generalisations. We define a special transformation with respect to two conic sections and a line intersecting the conics, and we prove several properties of the transformation that provide a solution to our generalised problem. Our main aim is to determine the invariant lines with respect to the transformation as a generalisation of the original IMO problem.
Keywords: Projective geometry tools, projectivity, perspectivity, invariant lines, generalisation.
MSC: 51M15; 51A05, 51N15.
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