好漂亮啊！
对于5#提出的问题：
我们可以容易得到一般解
O(r_1-r_2,0) , O_1( -r_1,0) ,O_2(r_2,0), O_3(r_3,y_3), O_4(-r_4,y_4)
x_3=r_3=r_4=(r_1*r_2)/(r_1+r_2)=-x_4, y_3=2*r_2*sqrt(r_1/(r_1+r_2))
y_4=2*r_1*sqrt(r_2/(r_1+r_2))
(1)对于r_(311),r_(312),r_(313),...,r_(31k)系列圆，其中k为大于1的正整数
(r_(31k)-r_2)^2+(y_(31k))^2=(r_(31k)+r_2)^2
(x_(31k)-r_2+r_1)^2+(y_(31k))^2=(r_2+r_1-r_(31k))^2
(x_(31k)-r_(31(k-1)))^2+(y_(31k)-y_(31(k-1)))^2=(r_(31k)+r_(31(k-1)))^2
特别对于r_(311),x_(311),y_(311) 的计算
(r_(311)-r_2)^2+(y_(311))^2=(r_(311)+r_2)^2
(x_(311)-r_2+r_1)^2+(y_(311))^2=(r_2+r_1-r_(311))^2
(x_(311)-r_3)^2+(y_(311)-y_3)^2=(r_(311)+r_3)^2
(2)对于r_(321),r_(322),r_(323),...,r_(32k)系列圆，其中k为大于1的正整数
(r_(32k)-r_2)^2+(y_(32k))^2=(r_(32k)+r_2)^2
(r_(32k)-r_(32(k-1)))^2+(y_(32k)-y_(32(k-1)))^2=(r_(32k)+r_(32(k-1)))^2
r_(32k)=x_(32k)
特别对于r_(321),x_(321),y_(321) 的计算
(r_(321)-r_2)^2+(y_(321))^2=(r_(321)+r_2)^2
(r_(321)-r_3)^2+(y_(321)-y_3)^2=(r_(321)+r_3)^2
r_(321)=x_(321)
(3)对于r_(331),r_(332),r_(333),...,r_(33k)系列圆，其中k为大于1的正整数
(r_(33k)-r_2)^2+(y_(33k))^2=(r_(33k)+r_2)^2
(x_(33k)-r_2+r_1)^2+(y_(33k))^2=(r_2+r_1-r_(33k))^2
x_(33k)=r_(33k)
特别对于r_(331),x_(331),y_(331) 的计算
(r_(331)-r_2)^2+(y_(331))^2=(r_(331)+r_2)^2
(x_(331)-r_2+r_1)^2+(y_(331))^2=(r_2+r_1-r_(331))^2
x_(331)=r_(331)
(4)对于r_(411),r_(412),r_(413),...,r_(41k)系列圆，其中k为大于1的正整数
(x_(41k)+r_1)^2+(y_(41k))^2=(r_(41k)+r_1)^2
(x_(41k)-r_2+r_1)^2+(y_(41k))^2=(r_2+r_1-r_(41k))^2
(x_(41k)-r_(41(k-1)))^2+(y_(41k)-y_(41(k-1)))^2=(r_(41k)+r_(41(k-1)))^2
特别对于r_(411),x_(411),y_(411) 的计算
(x_(411)+r_1)^2+(y_(411))^2=(r_(411)+r_1)^2
(x_(411)-r_2+r_1)^2+(y_(411))^2=(r_2+r_1-r_(411))^2
(x_(411)-r_4)^2+(y_(411)-y_4)^2=(r_(411)+r_4)^2
(5)对于r_(421),r_(422),r_(423),...,r_(42k)系列圆，其中k为大于1的正整数
(x_(42k)+r_1)^2+(y_(42k))^2=(r_(42k)+r_1)^2
(x_(42k)-r_(42(k-1)))^2+(y_(42k)-y_(42(k-1)))^2=(r_(42k)+r_(42(k-1)))^2
x_(42k)=-r_(42k)
特别对于r_(421),x_(421),y_(421) 的计算
(x_(421)+r_1)^2+(y_(421))^2=(r_(421)+r_1)^2
(x_(421)-r_4)^2+(y_(421)-y_4)^2=(r_(421)+r_4)^2
x_(421)=-r_(421)
(6)对于r_(431),r_(432),r_(433),...,r_(43k)系列圆，其中k为大于1的正整数
(x_(43k)+r_1)^2+(y_(43k))^2=(r_(43k)+r_1)^2
(x_(43k)-r_2+r_1)^2+(y_(43k))^2=(r_2+r_1-r_(43k))^2
x_(43k)=-r_(43k)
特别对于r_(431),x_(431),y_(431) 的计算
(x_(431)+r_1)^2+(y_(431))^2=(r_(431)+r_1)^2
(x_(431)-r_2+r_1)^2+(y_(431))^2=(r_2+r_1-r_(431))^2
x_(431)=-r_(431)
学习一下。
有人有些东西研究的真深:M:
数学星空 发表于 2012-4-9 22:01
hujunhua 发表于 2012-4-10 02:59
\(R_3=R_4={R_1R_2}/R\)
以点\(A\)为反演中心\(AB \cdot AC\)为反演幂作反演变换，(\(\odot O_i\)为\({c_i}\))则
\[{c_1} \to {l_1},{l_1} \to {c_1},{c_2} \to {l_2},{c_4} \to {c_4^\prime }, D \to P^\prime \]
\(A\),\(D\),\(P^\prime\)共线,\(A\),\(O_4\),\(O_4\)共线
\(c_4\)与\(c_1\),\(c_2\),\(l_1\)相切\(\Rightarrow \)\(c_4\)与\(l_1\),\(l_2\),\(c_1\)相切\(\Rightarrow \)\(R_{C_4^\prime}=R_{C_3}=R_3\)
\(P\)是\(c_2\)与\(c_4\)内位似中心\(\Rightarrow \)\(A\),\(P\),\(P^\prime\)共线\(\Rightarrow \)\(A\),\(P\),\(P^\prime\)共线
\(O_1P^\prime//O_2P\),\(A\)是\(c_1\)与\(c_2\)的位似中心
\[\Rightarrow \frac{PO_4}{PO_2}=\frac{P^\prime O_4^\prime}{P^\prime O_1}\]
即\(\frac{R_4}{R_2}=\frac{R_{C_4^\prime}}{R1}=\frac{R_3}{R_1}\)
\[\Rightarrow R_4=\frac{R_2R_3}{R_1}\]