On converses of Napoleon's theorem and a modified shape function.
On converses of Napoleon's theorem and a modified shape function.
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拿破仑定理的逆命题和修正形函数。
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发表时间：
2006
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通讯作者：
M. Spirova
中科院分区：
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作者：
Mowaffaq Hajja;H. Martini;M. Spirova
非简并（正向）三角形 ABC 的（负）托里切利三角形 T1(ABC) 被定义为三角形 A1B1C1，其中 ABC1、BCA1 和 CAB1 是在 ABC 的边上向外绘制的等边三角形。众所周知，并非每个三角形都是某个初始三角形的托里拆利三角形，并且非托里拆利三角形的三角形在[28]中进行了表征。在本文中表明，通过扩展 T1 的定义以包含退化三角形，映射 T1 变为双射，并且每个三角形都是唯一三角形的托里拆利三角形。还表明 T1 具有平滑特性，即对于任何初始三角形，迭代操作 T1 的过程在形状上收敛到等边三角形。对于内部直立的等边三角形也得到了类似的陈述，并且证明给出了琼·莱斯特形状函数的稍微修改的形式，预计该形式在其他情况下也有用。得出了与由 ABC1、BCA1 和 CAB1 创建的配置产生的各种三角形相关的几个进一步结果。这些指的是 Brocard 角度、透视属性和（定向）区域。 *第一作者得到了耶尔穆克大学的研究资助 0138-4821/93 $ 2.50 c © 2006 Heldermann Verlag 364 M. Hajja 等人：论拿破仑定理的逆向。 。 。
The (negative) Torricelli triangle T1(ABC) of a nondegenerate (positively oriented) triangle ABC is defined to be the triangle A1B1C1, where ABC1, BCA1, and CAB1 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 [28]. In the present article it is shown that, by extending the definition of T1 such that degenerate triangles are included, the mapping T1 becomes bijective and every triangle is then the Torricelli triangle of a unique triangle. It is also shown that T1 has the smoothing property, i.e., that the process of iterating the operations T1 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 ABC1, BCA1, and CAB1 are derived. These refer to Brocard angles, perspectivity properties, and (oriented) areas. ∗The first named author was supported by a research grant from Yarmouk University 0138-4821/93 $ 2.50 c © 2006 Heldermann Verlag 364 M. Hajja et al.: On Converses of Napoleon’s Theorem. . .