Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 51, No. 1, pp. 171-190 (2010)

Previous Article

Next Article

Contents of this Issue

Other Issues

ELibM Journals

ELibM Home



Generalized Napoleon and Torricelli transformations and their iterations

Mowaffaq Hajja

Department of Mathematics, Yarmouk University, Irbid, Jordan, e-mail:\quad

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

Full text of the article:

Electronic version published on: 27 Jan 2010. This page was last modified: 28 Jan 2013.

© 2010 Heldermann Verlag
© 2010–2013 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition