成都市1963年中学数学竞赛高三第二试中的第1题为“设三个复数z_1、z_2、z_3满足关系式:|z_1|=|z_2|=|z_3|与z_1+z_2+z_3=0,试证这三个复数在复平面上所表示的点是正三角形的顶点。”我们认为这是一道比较好的题。特别是结合现在的全国统编教材,对复数知识将是一次综合应用。在学生练习的基础上,我们总结了以下三种解法: 方法一: 设这三个复数在复平面上所表示的点 The first question in the second year of the high school math competition in Chengdu in 1963 was “Set three complex numbers z_1, z_2, z_3 to satisfy the relation: |z_1|=|z_2|=|z_3| and z_1+z_2+z_3=0 Try to prove that the points represented by the three complex numbers on the complex plane are the vertices of the equilateral triangle. ”We think this is a good question. In particular, combined with the current national textbooks, the complex knowledge will be a comprehensive application. Based on the students’ practice, we have summarized the following three solutions: Method 1: Let these three complex numbers be represented on the complex plane