Generalized Napoleon and Torricelli Transformations and Their Iterations

For given triangles T = (A,B,C) and D = (X, Y, Z), the D-Napoleon and D-Torricelli triangles ND(T ) and TD(T ) of a triangle T = (A,B,C) are the triangles A′B′C ′ and A∗B∗C∗, where ABC ′, BCA′, CAB′, A∗BC, AB∗C, ABC∗ are similar to D. In this paper it is shown that the iteration N n D(T ) either terminates or converges (in shape) to an equilateral triangle, and that the iteration T n D (T ) either terminates or converges to a triangle whose shape depends only on D. It is also shown that if A◦, B◦, C◦, A, B, C are the centroids of the triangles ABC ′, BCA′, CAB′, A∗BC, AB∗C, ABC∗, respectively, then the shape of A◦B◦C◦ depends on both shapes of T and D, while the shape of ABC depends only on that of D and, unexpectedly, equals the limiting shape of the iteration T n D (T ).