On converses of Napoleon's theorem and a modified shape function

Mowaffaq Hajja, Horst Martini and Margarita Spirova

Abstract: The (negative) Torricelli triangle $\{\mathcal T}_1 (ABC)$ of a non-degenerate (positively oriented) triangle $ABC$ is defined to be the triangle $A_1B_1C_1$, where $ABC_1$, $BCA_1$, and $CAB_1$ are the equilateral triangles drawn outwardly on the sides of $ABC$. It is known that not every triangle is the Torricelli triangle of some initial triangle, and triangles that are not Torricelli triangles are characterized in [S]. In the present article it is shown that, by extending the definition of ${\mathcal T}_1$ such that degenerate triangles are included, the mapping ${\mathcal T}_1$ becomes bijective and every triangle is then the Torricelli triangle of a unique triangle. It is also shown that ${\mathcal T}_1$ has the smoothing property, i.e., that the process of iterating the operations ${\mathval T}_1$ converges, in shape, to an equilateral triangle for any initial triangle. Analogous statements are obtained for internally erected equilateral triangles, and the proofs give rise to a slightly modified form of June Lester's shape function which is expected to be useful also in other contexts. Several further results pertaining to the various triangles that arise from the configuration created by $ABC_1$, $BCA_1$, and $CAB_1$ are derived. These refer to Brocard angles, perspectivity properties, and (oriented) areas.

\smallskip

[S] Sakmar, I. A.: \emph{Problem E3257}. Amer. Math. Monthly {\bf 95} (1988), 259; solution: ibid {\bf 98} (1991), 55--57.

Keywords: anti-medial triangle, anti-similar triangle, area, Brocard angle, complementary triangle, complex number, Gaussian plane, medial triangle, Möbius transformation, Napoleon configuration, Napoleon triangle, oriented area, perspectivity, shape function, similar triangles, smoothing iteration, Torricelli configuration, Torricelli triangle

Full text of the article:

• Compressed PostScript file (629 kilobytes)
• PDF file (276 kilobytes)