鞋匠刀问题

gxqcn

好漂亮啊！

数学星空

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$ $(x_(311)-r_3)^2+(y_(311)-y_3)^2=(r_(311)+r_3)^2$

数学星空

(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)$

数学星空

(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) $

数学星空

(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$ $(x_(411)-r_4)^2+(y_(411)-y_4)^2=(r_(411)+r_4)^2$

数学星空

(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)$

数学星空

(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$ $x_(431)=-r_(431)$

winxos

学习一下。

mathtime

有人有些东西研究的真深

葡萄糖

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\(\frac{R_4}{R_2}=\frac{R_{C_4^\prime}}{R1}=\frac{R_3}{R_1}\)
\[\Rightarrow R_4=\frac{R_2R_3}{R_1}\]