Generalized Napoleon and Torricelli transformations and their iterations

Mowaffaq Hajja

Abstract: 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, Fermat-Torricelli point, generalized Napoleon configuration, generalized Napoleon triangle, generalized Torricelli configuration, generalized Torricelli triangle, Möbius transformation, shape convergence, shape function, similar triangles, smoothing iteration

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