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读了贵刊82年第4期中《三复数组成正三角形的充要条件教学探讨》一文很受启发,王先俊同志采用循序渐进的编排方法,对命题“z_1,z_2,z_3组成一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”作了多种证明,考虑到该命题很有实用价值,我想在王文的基础上再补充讲授运用该命题来方便地解决一类涉及正三角形顶点的计算,证明和求轨迹等问题,以提高学生对此命题的认识。例1.已知一个正三角形的两个顶点分别是A=1,B=2十i,求表示第三个顶点C的复数。解:据命题性质有
After reading the article in the 4th issue of the Journal of the 3rd Issue of the Teaching of Necessary and Sufficient Conditions for the Formation of the Trigonometric and Polygonal Triangles, Comrade Wang Xianjun adopts a step-by-step method of arranging the proposition “z_1, z_2, z_3 to form an equilateral triangle. The necessary and sufficient condition for the vertices is that they are suitable for the equation z_1~2+z_2~2+z_3~2=z_2z_3+z_3z_1+z_1z_2”. Various proofs are given. Considering that the proposition has practical value, I want to use it in Wang Wen’s On the basis of this, it is added that this proposition is used to conveniently solve a class of calculations involving the vertices of an equilateral triangle, and to prove and seek the trajectory, in order to improve students’ understanding of this proposition. Example 1. It is known that the two vertices of an equilateral triangle are A=1 and B=2 and i, respectively, to represent the complex number of the third vertex C. Solution: According to the nature of the proposition