On the Iterated Cevian Triangle in Finite Euclidean Space

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.