The dynamical system of iterated Cevian Tribbles
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Ceva's Theorem gives a necessary and sufficient condition for three lines through the vertices of a triangle to intersect at a single point. We investigate what happens when that condition is not met, which means the three lines form a triangle inside the original (called a Tribble), and the process is iterated. By Cantor's Intersection Theorem, we know that the Tribbles will converge to a point within the initial triangle as long as the side lengths of the Tribbles go to zero. We consider different ways to iterate this process. We establish an nth term test for convergence when the Cevian ratios are deterministic sequences. We prove that when the Cevian ratios used to iterate are chosen at random, the Tribbles always converge. We initiate study on the distribution of limit points. With the aid of a simulation in MATLAB that produces graphical plots for the numerical estimations of Tribble limit points, we begin to visualize and describe the distribution of limit points.