单位圆上四个点 直线的交点

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Formulas for Complex Numbers in Geometry
当 A、B、C、D 四个点位于单位圆上时，直线 AB 与 CD 的交点 E 由下式给出：
$$
\bar{e}=\frac{a+b-c-d}{a b-c d} \quad \text { or } \quad e=\frac{a b c+a b d-a c d-b c d}{a b-c d} .
$$

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从 $\frac{q-a}{c-a}\inR$ 得到$$(c-a)(q^*-a^*)=(c^*-a^*)(q-a)$$
代入 $a^*=a^{-1},c^* =c^{-1}$ 得到\begin{equation}q+ ac q^* = a+c\label1\end{equation}类似地得到\begin{equation}q+ bd q^* = b+d\label2\end{equation}\eqref{1}、\eqref{2}组成关于 $q,q^*$ 的二元一次方程组，解得 $q,q^*$

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2018年剑桥大学本科入学测试题第六题 | TeX文件

The distinct points $A$, $B$, $C$ and $D$ lie, in anticlockwise order, on the circle of unit radius with centre at the origin (so that, for example, $aa^* =1$).
The lines $AC$ and $BD$ meet at $Q$.
Show that
\[
(ac-bd)q^* = (a+c)-(b+d)
\,,
\]
where $b$ and $d$  are complex numbers represented by the points $B$ and $D$ respectively, and show further that
\[
(ac-bd)
(q+q^*) =
(a-b)(1+cd) +(c-d)(1+ab)
\,.
\]

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hbghlyj 发表于 2024-11-3 10:06
Formulas for Complex Numbers in Geometry

文件中的前一段为：

If $a$ and $b$ are on the unit circle, the equation of secant line $A B$ is $z=a+b-a b \bar{z}$.
If $a=b$, this is the equation of the tangent line at $a$.

这里的 $z=a+b-a b \bar{z}$ 就是\eqref{1}.