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全日制十年制高中数学第三册有一道复习题(习题三第18题):求证三个复数z_1,z_2,z_3组成一个等边三角形的三个顶点的充要条件是它们适合等式z_(1~2)+z_(2~2)+z_(3~2)=z_2z_3+z_3z_1+z_1z_2。对于学生来说,有一定的难度,就是许多初教者,往往讲解起来也有困难。笔者采用如下几种证法,较之《教参》中的证明简洁易懂,效果较好,兹介绍如下。为叙述方便,将“必要性”和“充分性”分别集中。
There is a review question in the third-year full-year ten-year high school mathematics book (question 3, item 18): The necessary and sufficient condition for the verification of the three complex numbers z_1, z_2, z_3 and the three vertices of an equilateral triangle is that they fit the equation z_ (1~2)+z_(2~2)+z_(3~2)=z_2z_3+z_3z_1+z_1z_2. For students, there is a certain degree of difficulty. Many beginners often have difficulties in explaining. The author uses the following types of verification methods, which is simpler and easier to understand than the proof in the “Guidance Reference”. For the sake of narrative convenience, “necessity” and “adequacy” are separately concentrated.