Coincidence of Centers for Scalene Triangles
Coincidence of Centers for Scalene Triangles
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不等边三角形中心的重合
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
Mowaffaq Hajja
中科院分区：
文献类型：
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作者：
S. Abu;Mowaffaq Hajja
一个中心函数是一个函数Z，它以对称的方式给欧几里得平面E中的每个三角形T分配一个点Z（T），并且考虑等距和膨胀。一个中心函数族F称为完备的，如果对于每个不等边三角形ABC和它平面上的每个点P，存在Z∈ F使得Z（ABC）= P。本文给出了连续三角形中心函数的完全分离族的简单例子。关于没有两个不同的中心函数可以在一个不等边三角形上重合的印象，我们证明了对于每个中心函数Z和每个不等边三角形T，存在另一个简单类型的中心函数Z，使得Z（T）= Z（T）。
A center function is a function Z that assigns to every triangle T in a Euclidean plane E a point Z(T) in E in a manner that is symmetric and that respects isometries and dilations. A family F of center functions is said to be complete if for every scalene triangle ABC and every point P in its plane, there is Z∈ F such that Z(ABC )= P. It is said to be separating if no two center functions in F coincide for any scalene triangle. In this note, we give simple examples of complete separating families of continuous triangle center functions. Regarding the impression that no two different center functions can coincide on a scalene triangle, we show that for every center function Z and every scalene triangle T , there is another center function Z, of a simple type, such that Z(T )= Z(T).