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Abstract
This paper investigates the iterative process of constructing Cevian triangles in a finite Euclidean plane. First, we prove that starting from an initial triangle, the process of iteratively taking Cevian triangles converges to a unique fixed point. Second, we show this convergence process is surjective onto the interior of the triangle; that is, for any target point in the interior, one can find an initial point whose sequence of iterated Cevian triangles converges to that target. Finally, we examine the limiting configuration of an infinite iteration and characterize a novel property intrinsic to the discrete nature of the finite geometric space, setting it apart from the classical real Euclidean case.
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References
- K. Seebach, “Ceva-dreiecke,†Elem. Math, vol. 42, pp. 132—-139, 1987.
- H. Mowaffaq, “The arbitrariness of cevian triangle,†The American mathematical monthly, vol. 113, no. 5, pp. 443–447, 2006. https://doi.org/10.2307/27641951.
- B. Hvala, “Cevian cousins of a triangle centroid,†Journal for Geometry and Graphics, vol. 19, no. 2, pp. 211–218, 2015. https://www.heldermann-verlag.de/jgg/jgg19/j19h2hval.pdf.
- Y. N. Aliyev, “Inequalities about the area bounded by three cevian lines of a triangle,†Elem. Math., vol. 80, no. 4, pp. 137—-148, 2025. https://doi.org/10.4171/EM/540.
- E. Carroll, A. P. Ghosh, X. H. Nguyen, and A. Roitershtein, “Iterated routh’s triangles,†Journal for Geometry and Graphics, vol. 21, no. 2, pp. 153–168, 2017. https://www.heldermann-verlag.de/jgg/jgg21/j21h2carr.pdf.
- F. Brunck, “Iterated medial triangle subdivision in surfaces of constant curvature,†Discrete Comput Geom, vol. 70, pp. 1059–1089, 2023. https://doi.org/10.1007/s00454-023-00500-5.
- R. T. Rockafellar, Convex Analysis. Princeton University Press, 1970. https://doi.org/10.1017/S0013091500010142.
- I. E. Leonard and J. E. Lewis, Geometry of convex sets. Wiley, 2016.
References
K. Seebach, “Ceva-dreiecke,†Elem. Math, vol. 42, pp. 132—-139, 1987.
H. Mowaffaq, “The arbitrariness of cevian triangle,†The American mathematical monthly, vol. 113, no. 5, pp. 443–447, 2006. https://doi.org/10.2307/27641951.
B. Hvala, “Cevian cousins of a triangle centroid,†Journal for Geometry and Graphics, vol. 19, no. 2, pp. 211–218, 2015. https://www.heldermann-verlag.de/jgg/jgg19/j19h2hval.pdf.
Y. N. Aliyev, “Inequalities about the area bounded by three cevian lines of a triangle,†Elem. Math., vol. 80, no. 4, pp. 137—-148, 2025. https://doi.org/10.4171/EM/540.
E. Carroll, A. P. Ghosh, X. H. Nguyen, and A. Roitershtein, “Iterated routh’s triangles,†Journal for Geometry and Graphics, vol. 21, no. 2, pp. 153–168, 2017. https://www.heldermann-verlag.de/jgg/jgg21/j21h2carr.pdf.
F. Brunck, “Iterated medial triangle subdivision in surfaces of constant curvature,†Discrete Comput Geom, vol. 70, pp. 1059–1089, 2023. https://doi.org/10.1007/s00454-023-00500-5.
R. T. Rockafellar, Convex Analysis. Princeton University Press, 1970. https://doi.org/10.1017/S0013091500010142.
I. E. Leonard and J. E. Lewis, Geometry of convex sets. Wiley, 2016.