圆幂之比为定值 共轴圆组

hbghlyj

设一点在共轴圆组的一个圆上移动，则这点关于这组中另两个圆的幂的比是一个定值，即这点所在圆的圆心到另两个圆圆心的距离的比.
证明 1: 见《近欧》§114
设这三个圆为 $c, c_1, c_2$; $P$ 为 $c$ 上任意一点, $P$ 到根轴的垂线为 $P Q$, $P$ 关于圆 $c$ 的幂为 $P(c)$, 则
$$
P(c)=0, P\left(c_1\right)=2 \overline{P Q} \cdot \overline{C C_1}, P\left(c_2\right)=2 \overline{P Q} \cdot \overline{C C_2} \text {. }
$$
因此立即得
$$
\frac{P\left(c_1\right)}{P\left(c_2\right)}=\frac{\overline{C C_1}}{\overline{C C_2}},
$$
证明 2:
共轴圆组$\Gamma$是经过两个点$A,B(A≠B)$(坐标是实数或共轭虚数)的所有圆的集合.
设$\{(x,y):P(x,y)=0\}∈\Gamma$, 则$P(A)=P(B)=0,Q(A)=Q(B)=0$, 所以$(λP+Q)(A)=(λP+Q)(B)=0$.
而$(λP+Q)(x,y)=0$是一个圆, 所以$\{(x,y):(λP+Q)(x,y)=0\}∈\Gamma$, 所以$\{\{(x,y):(λP+Q)(x,y)=0\}:λ∈\Bbb R\}\subseteq\Gamma$.
反过来对任意点$C(≠A,B)$, 过$A,B,C$的圆是唯一的, 使$(λP+Q)(C)=0$的$λ=\frac{-Q(C)}{P(C)}$也是唯一的, 所以$\{\{(x,y):(λP+Q)(x,y)=0\}:λ∈\Bbb R\}=\Gamma$.

hbghlyj

共轴圆组$\Gamma$是经过两个点$A,B(A≠B)$(坐标是实数或共轭虚数)的所有圆的集合.

比如$(x-1)^2+y^2=0$和$(x+1)^2+y^2=0$和$x^2=0$是共轴圆, 它们都经过$(0,\pm i)$

Continuity Principle
... there will be the same number of solutions in all cases, although some solutions may be imaginary.
For example, two circles intersect in two points, so it can be stated that every two circles intersect in two points, although the points may be imaginary or may coincide.

hbghlyj

关于“共轴圆组”又见
Cut the knot
Wikipedia–Apollonian circles
MathWorld–Coaxal Circles
2013AAGChapter3.pdf
Pencils of circles or coaxal circles–Geometrikon
Harmonic ranges in a coaxal system of circles
D. Pedoe, Circles: A Mathematical View, MAA, 1995

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H. A. Priestley - Introduction to Complex Analysis (2003) page 21
2.12 Coaxal circles. For distinct fixed points $\alpha$ and $\beta$ in $\mathbb{C}$, we have, as $\lambda$ and $\mu$ vary, two families of circles:

• $C_1(\alpha, \beta)$: circles
  $$
  \left|\frac{z-\alpha}{z-\beta}\right|=\lambda,
  $$
  having $\alpha$ and $\beta$ as inverse points. Here $\lambda=1$ is allowed, so that the straight line bisecting the segment $[\alpha, \beta]$ is included.
• $C_2(\alpha, \beta)$: circles
  $$
  \arg \left(\frac{z-\alpha}{z-\beta}\right)=\left\{\begin{array}{l}
  \mu \\
  -(\pi-\mu)
  \end{array} \pmod{2 \pi},\right.
  $$
  through $\alpha$ and $\beta$.

Traditionally, each of the families $C_1(\alpha, \beta)$ and $C_2(\alpha, \beta)$ is said to form a coaxal system. Coaxal systems of circles have interesting geometric properties. For example, any member of $C_1(\alpha, \beta)$ cuts any member of $C_2(\alpha, \beta)$ orthogonally.

由此可见 Möbius transformation $f(z)=\frac{z-\alpha}{z-\beta}$ 把$C_1(α,β),C_2(α,β)$映射到以0为中心的圆和经过0的直线.
当$\beta=1$时$f=f^{-1}$
而$\exp$将平行于坐标轴的直线映射到以0为中心的圆和经过0的直线.
将$f^{-1}$与$\exp$复合得到$\tan$ [准确来说是$\tan(z)=i\frac{\exp(2iz)+1}{\exp(2iz)-1}$]
见Wolfram functions的条目Tan

Images of a square grid of size $[-\pi,\pi]\times[-\pi,\pi]$ under the (conformal) map $z\mapsto\tan z$. Asymptotically, points are mapped to the two points $\pm i$.