複數$a\bar a=b\bar b=c\bar c$

hbghlyj

GTM123. Numbers (2nd edition), edited by J. H. Ewing. p72
For all $a,b,c,d\inC$ with $a\bar a=b\bar b=c\bar c$ we have
$$(a-b)(c-d)(\bar{a}-\bar{d})(\bar{c}-\bar{b})+i(c \bar{c}-d \bar{d}) \operatorname{Im}(c \bar{b}-c \bar{a}-a \bar{b}) \in \mathbf{R}$$

hejoseph

这个只要硬算也不难的，设
\begin{align*}
a&=r(\cos t+\mathrm{i}\sin t)\\
b&=r(\cos u+\mathrm{i}\sin u)\\
c&=r(\cos v+\mathrm{i}\sin v)\\
d&=x+y\mathrm{i}
\end{align*}
那么
\begin{align*}
&(a-b)(c-d)(\bar{a}-\bar{d})(\bar{c}-\bar{b})+\mathrm{i}(c \bar{c}-d \bar{d}) \operatorname{Im}(c \bar{b}-c \bar{a}-a \bar{b})\\
={}&{-}4r^2\sin\frac{t-u}{2}\sin\frac{u-v}{2}\left((r^2+x^2+y^2)\cos\frac{t-v}{2}-2r\left(x\cos\frac{t+v}{2}+y\sin\frac{t+v}{2}\right)\right)
\end{align*}