回覆 8# 5pn3gp6 的帖子
這題跟今年師大附中的填充第 7 題一樣
原題出自 1999 年大陸的全國高中數學聯賽
z_1、z_2、z_3 應該非完全相異
為打字方便,把 z_1、z_2、z_3 分別以 p、q、r 表示,其共軛複數分別是 p'、q'、r'
p/q + q/r + r/p = (p/q + q/r + r/p)' = (p/q)' + (q/r)' + (r/p)' = p'/q' + q'/r' + r'/p'
|p| = |q| = |r| = 1
p' = 1/p、q' = 1/q、r' = 1/r 代入上式可得
p/q + q/r + r/p = q/p + r/q + p/r
同乘以 pqr
p^2r + q^2p + r^2q = q^2r + r^2p + p^2q
(p - q)(q - r)(r - p) = 0
p = q 或 q = r 或 r = p
若 p = q
1 + p/r + r/p = 1
r/p = ±i
|ap + bq + cr| = |p||a + b ± ci| = √[(a + b)^2 + c^2]
同理
若 q = r,|ap + bq + cr| = √[(b + c)^2 + a^2]
若 r = p,|ap + bq + cr| = √[(c + a)^2 + b^2]