Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 51, No. 1, pp. 171190 (2010) 

Generalized Napoleon and Torricelli transformations and their iterationsMowaffaq HajjaDepartment of Mathematics, Yarmouk University, Irbid, Jordan, email: mowhajja@yahoo.com\quad mhajja@yu.edu.joAbstract: For given triangles $T=(A,B,C)$ and $D=(X,Y,Z)$, the $D$Napoleon and $D$Torricelli triangles $\NAP_D(T)$ and $\TOR_D (T)$ 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 $\NAP_D^n(T)$ either terminates or converges (in shape) to an equilateral triangle, and that the iteration $\TOR_D^n(T)$ either terminates or converges to a triangle whose shape depends only on $D$. It is also shown that if $A^{\circ}$, $B^{\circ}$, $C^{\circ}$, $A^{\cc}$, $B^{\cc}$, $C^{\cc}$ are the centroids of the triangles $ABC'$, $BCA'$, $CAB'$, $A^*BC$, $AB^*C$, $ABC^*$, respectively, then the shape of $A^{\circ} B^{\circ} C^{\circ}$ depends on both shapes of $T$ and $D$, while the shape of $A^{\cc} B^{\cc} C^{\cc}$ depends only on that of $D$ and, unexpectedly, equals the limiting shape of the iteration $\TOR_D^n(T)$. Keywords: centroids, (plane of) complex numbers, FermatTorricelli point, generalized Napoleon configuration, generalized Napoleon triangle, generalized Torricelli configuration, generalized Torricelli triangle, Möbius transformation, shape convergence, shape function, similar triangles, smoothing iteration Full text of the article:
Electronic version published on: 27 Jan 2010. This page was last modified: 28 Jan 2013.
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