N边形外接与内切圆半径问题

数学星空

  在 http://bbs.emath.ac.cn/forum.php ... 10&fromuid=1455    根据mathe 提供的解答

于是得出
${2a^2b^2r^2}/{a^2-x_0^2+b^2-y_0^2+r^2}+{(a^2-x_0^2+b^2-y_0^2+r^2)^2}/4=a^2b^2-a^2y_0^2+a^2r^2-b^2x_0^2+b^2r^2$
其中圆的半径为r
另外我们可以采用一个方向无关的记号，设$d^2=x_0^2+y_0^2$,d为$(x0,y0)$到圆心的距离,$t$为向量$(a,b)$和$(x_0,y_0)$的夹角，于是可以写成
${2a^2b^2r^2}/{a^2+b^2+r^2-d^2}+{(a^2+b^2+r^2-d^2)^2}/4=a^2b^2+(a^2+b^2)(r^2-d^2sin^2(t))$
或者
$(r^2+a^2+b^2-d^2)^3-4(r^2+a^2+b^2-d^2)((a^2+b^2)(r^2-d^2sin^2(t))+a^2b^2)+8a^2b^2r^2=0$
于是t=0,可以得出前面得出$x_0=0$或$y_0=0$的情况
$(r^2+a^2+b^2-d^2)^3-4(r^2+a^2+b^2-d^2)((a^2+b^2)r^2+a^2b^2)+8a^2b^2r^2=0$
而d=0，也就是中心重叠可以变化为
$(r^2+a^2+b^2)^3-4(r^2+a^2+b^2)((a^2+b^2)r^2+a^2b^2)+8a^2b^2r^2=0$

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中有一步代换：\(a^2y_0^2+b^2x_0^2=(a^2+b^2)d^2\sin(t)^2\)应该是有问题的。

我认为应该改为：\(a^2y_0^2+b^2x_0^2=d^2((a^2-b^2)\sin(t)^2+b^2)\)

因此上面后面部份有一句应修改为：

于是\(t=0\),可以得出前面得出$x_0=0$或$y_0=0$的情况

$(r^2+a^2+b^2-d^2)^3-4(r^2+a^2+b^2-d^2)((a^2+b^2)r^2-d^2b^2)+8a^2b^2r^2=0$

数学星空

根据  http://bbs.emath.ac.cn/forum.php ... 56&fromuid=1455

对于双圆的结果：外切于圆\(x^2+y^2=r^2\)，内接于圆 \((x-x_0)^2+(y-y_0)^2=R^2\)的n边形的存在条件为：

\(n=3\)时

\(R^2-2Rr-x_0^2-y_0^2=0\)

\(n=4\)时

\(R^4-2R^2r^2-2R^2x_0^2-2R^2y_0^2-2r^2x_0^2-2r^2y_0^2+x_0^4+2x_0^2y_0^2+y_0^4=0\)

\(n=5\)时

\(R^6+2R^5r-4R^4r^2-3R^4x_0^2-3R^4y_0^2-4R^3rx_0^2-4R^3ry_0^2+4R^2r^2x_0^2+4R^2r^2y_0^2+3R^2x_0^4+6R^2x_0^2y_0^2+3R^2y_0^4-8Rr^3x_0^2-8Rr^3y_0^2+2Rrx_0^4+4Rrx_0^2y_0^2+2Rry_0^4-x_0^6-3x_0^4y_0^2-3x_0^2y_0^4-y_0^6=0\)

\(n=6\)时

\(3R^8-4R^6r^2-12R^6x_0^2-12R^6y_0^2+4R^4r^2x_0^2+4R^4r^2y_0^2+18R^4x_0^4+36R^4x_0^2y_0^2+18R^4y_0^4-16R^2r^4x_0^2-16R^2r^4y_0^2+4R^2r^2x_0^4+8R^2r^2x_0^2y_0^2+4R^2r^2y_0^4-12R^2x_0^6-36R^2x_0^4y_0^2-36R^2x_0^2y_0^4-12R^2y_0^6-4r^2x_0^6-12r^2x_0^4y_0^2-12r^2x_0^2y_0^4-4r^2y_0^6+3x_0^8+12x_0^6y_0^2+18x_0^4y_0^4+12x_0^2y_0^6+3y_0^8=0\)

\(n=7\)时

\(R^{12}-4R^{11}r-4R^{10}r^2-6R^{10}x_0^2-6R^{10}y_0^2+8R^9r^3+20R^9rx_0^2+20R^9ry_0^2+16R^8r^2x_0^2+16R^8r^2y_0^2+15R^8x_0^4+30R^8x_0^2y_0^2+15R^8y_0^4-40R^7rx_0^4-80R^7rx_0^2y_0^2-40R^7ry_0^4-16R^6r^4x_0^2-16R^6r^4y_0^2-24R^6r^2x_0^4-48R^6r^2x_0^2y_0^2-24R^6r^2y_0^4-20R^6x_0^6-60R^6x_0^4y_0^2-60R^6x_0^2y_0^4-20R^6y_0^6-32R^5r^5x_0^2-32R^5r^5y_0^2-48R^5r^3x_0^4-96R^5r^3x_0^2y_0^2-48R^5r^3y_0^4+40R^5rx_0^6+120R^5rx_0^4y_0^2+120R^5rx_0^2y_0^4+40R^5ry_0^6+64R^4r^6x_0^2+64R^4r^6y_0^2+32R^4r^4x_0^4+64R^4r^4x_0^2y_0^2+32R^4r^4y_0^4+16R^4r^2x_0^6+48R^4r^2x_0^4y_0^2+48R^4r^2x_0^2y_0^4+16R^4r^2y_0^6+15R^4x_0^8+60R^4x_0^6y_0^2+90R^4x_0^4y_0^4+60R^4x_0^2y_0^6+15R^4y_0^8+64R^3r^3x_0^6+192R^3r^3x_0^4y_0^2+192R^3r^3x_0^2y_0^4+64R^3r^3y_0^6-20R^3rx_0^8-80R^3rx_0^6y_0^2-120R^3rx_0^4y_0^4-80R^3rx_0^2y_0^6-20R^3ry_0^8-16R^2r^4x_0^6-48R^2r^4x_0^4y_0^2-48R^2r^4x_0^2y_0^4-16R^2r^4y_0^6-4R^2r^2x_0^8-16R^2r^2x_0^6y_0^2-24R^2r^2x_0^4y_0^4-16R^2r^2x_0^2y_0^6-4R^2r^2y_0^8-6R^2x_0^{10}-30R^2x_0^8y_0^2-60R^2x_0^6y_0^4-60R^2x_0^4y_0^6-30R^2x_0^2y_0^8-6R^2y_0^{10}+32Rr^5x_0^6+96Rr^5x_0^4y_0^2+96Rr^5x_0^2y_0^4+32Rr^5y_0^6-24Rr^3x_0^8-96Rr^3x_0^6y_0^2-144Rr^3x_0^4y_0^4-96Rr^3x_0^2y_0^6-24Rr^3y_0^8+4Rrx_0^{10}+20Rrx_0^8y_0^2+40Rrx_0^6y_0^4+40Rrx_0^4y_0^6+20Rrx_0^2y_0^8+4Rry_0^{10}+x_0^{12}+6x_0^{10}y_0^2+15x_0^8y_0^4+20x_0^6y_0^6+15x_0^4y_0^8+6x_0^2y_0^{10}+y_0^{12}=0\)


\(n=8\)时

\(R^{16}-8R^{14}r^2-8R^{14}x_0^2-8R^{14}y_0^2+8R^{12}r^4+40R^{12}r^2x_0^2+40R^{12}r^2y_0^2+28R^{12}x_0^4+56R^{12}x_0^2y_0^2+28R^{12}y_0^4+48R^{10}r^4x_0^2+48R^{10}r^4y_0^2-72R^{10}r^2x_0^4-144R^{10}r^2x_0^2y_0^2-72R^{10}r^2y_0^4-56R^{10}x_0^6-168R^{10}x_0^4y_0^2-168R^{10}x_0^2y_0^4-56R^{10}y_0^6-128R^8r^6x_0^2-128R^8r^6y_0^2-264R^8r^4x_0^4-528R^8r^4x_0^2y_0^2-264R^8r^4y_0^4+40R^8r^2x_0^6+120R^8r^2x_0^4y_0^2+120R^8r^2x_0^2y_0^4+40R^8r^2y_0^6+70R^8x_0^8+280R^8x_0^6y_0^2+420R^8x_0^4y_0^4+280R^8x_0^2y_0^6+70R^8y_0^8+128R^6r^8x_0^2+128R^6r^8y_0^2+128R^6r^6x_0^4+256R^6r^6x_0^2y_0^2+128R^6r^6y_0^4+416R^6r^4x_0^6+1248R^6r^4x_0^4y_0^2+1248R^6r^4x_0^2y_0^4+416R^6r^4y_0^6+40R^6r^2x_0^8+160R^6r^2x_0^6y_0^2+240R^6r^2x_0^4y_0^4+160R^6r^2x_0^2y_0^6+40R^6r^2y_0^8-56R^6x_0^{10}-280R^6x_0^8y_0^2-560R^6x_0^6y_0^4-560R^6x_0^4y_0^6-280R^6x_0^2y_0^8-56R^6y_0^{10}+128R^4r^6x_0^6+384R^4r^6x_0^4y_0^2+384R^4r^6x_0^2y_0^4+128R^4r^6y_0^6-264R^4r^4x_0^8-1056R^4r^4x_0^6y_0^2-1584R^4r^4x_0^4y_0^4-1056R^4r^4x_0^2y_0^6-264R^4r^4y_0^8-72R^4r^2x_0^{10}-360R^4r^2x_0^8y_0^2-720R^4r^2x_0^6y_0^4-720R^4r^2x_0^4y_0^6-360R^4r^2x_0^2y_0^8-72R^4r^2y_0^{10}+28R^4x_0^{12}+168R^4x_0^{10}y_0^2+420R^4x_0^8y_0^4+560R^4x_0^6y_0^6+420R^4x_0^4y_0^8+168R^4x_0^2y_0^{10}+28R^4y_0^{12}+128R^2r^8x_0^6+384R^2r^8x_0^4y_0^2+384R^2r^8x_0^2y_0^4+128R^2r^8y_0^6-128R^2r^6x_0^8-512R^2r^6x_0^6y_0^2-768R^2r^6x_0^4y_0^4-512R^2r^6x_0^2y_0^6-128R^2r^6y_0^8+48R^2r^4x_0^{10}+240R^2r^4x_0^8y_0^2+480R^2r^4x_0^6y_0^4+480R^2r^4x_0^4y_0^6+240R^2r^4x_0^2y_0^8+48R^2r^4y_0^{10}+40R^2r^2x_0^{12}+240R^2r^2x_0^{10}y_0^2+600R^2r^2x_0^8y_0^4+800R^2r^2x_0^6y_0^6+600R^2r^2x_0^4y_0^8+240R^2r^2x_0^2y_0^{10}+40R^2r^2y_0^{12}-8R^2x_0^{14}-56R^2x_0^{12}y_0^2-168R^2x_0^{10}y_0^4-280R^2x_0^8y_0^6-280R^2x_0^6y_0^8-168R^2x_0^4y_0^{10}-56R^2x_0^2y_0^{12}-8R^2y_0^{14}+8r^4x_0^{12}+48r^4x_0^{10}y_0^2+120r^4x_0^8y_0^4+160r^4x_0^6y_0^6+120r^4x_0^4y_0^8+48r^4x_0^2y_0^{10}+8r^4y_0^{12}-8r^2x_0^{14}-56r^2x_0^{12}y_0^2-168r^2x_0^{10}y_0^4-280r^2x_0^8y_0^6-280r^2x_0^6y_0^8-168r^2x_0^4y_0^{10}-56r^2x_0^2y_0^{12}-8r^2y_0^{14}+x_0^{16}+8x_0^{14}y_0^2+28x_0^{12}y_0^4+56x_0^{10}y_0^6+70x_0^8y_0^8+56x_0^6y_0^{10}+28x_0^4y_0^{12}+8x_0^2y_0^{14}+y_0^{16}=0\)


\(n=9\)时

1. R^{18}+6R^{17}r-9R^{16}x_0^2-9R^{16}y_0^2-8R^{15}r^3-48R^{15}rx_0^2-48R^{15}ry_0^2+36R^{14}x_0^4+72R^{14}x_0^2y_0^2+36R^{14}y_0^4-8R^{13}r^3x_0^2-8R^{13}r^3y_0^2+168R^{13}rx_0^4+336R^{13}rx_0^2y_0^2+168R^{13}ry_0^4-96R^{12}r^4x_0^2-96R^{12}r^4y_0^2-84R^{12}x_0^6-252R^{12}x_0^4y_0^2-252R^{12}x_0^2y_0^4-84R^{12}y_0^6+32R^{11}r^5x_0^2+32R^{11}r^5y_0^2+216R^{11}r^3x_0^4+432R^{11}r^3x_0^2y_0^2+216R^{11}r^3y_0^4-336R^{11}rx_0^6-1008R^{11}rx_0^4y_0^2-1008R^{11}rx_0^2y_0^4-336R^{11}ry_0^6+256R^{10}r^6x_0^2+256R^{10}r^6y_0^2+480R^{10}r^4x_0^4+960R^{10}r^4x_0^2y_0^2+480R^{10}r^4y_0^4+126R^{10}x_0^8+504R^{10}x_0^6y_0^2+756R^{10}x_0^4y_0^4+504R^{10}x_0^2y_0^6+126R^{10}y_0^8+32R^9r^5x_0^4+64R^9r^5x_0^2y_0^2+32R^9r^5y_0^4-680R^9r^3x_0^6-2040R^9r^3x_0^4y_0^2-2040R^9r^3x_0^2y_0^4-680R^9r^3y_0^6+420R^9rx_0^8+1680R^9rx_0^6y_0^2+2520R^9rx_0^4y_0^4+1680R^9rx_0^2y_0^6+420R^9ry_0^8-256R^8r^8x_0^2-256R^8r^8y_0^2-512R^8r^6x_0^4-1024R^8r^6x_0^2y_0^2-512R^8r^6y_0^4-960R^8r^4x_0^6-2880R^8r^4x_0^4y_0^2-2880R^8r^4x_0^2y_0^4-960R^8r^4y_0^6-126R^8x_0^{10}-630R^8x_0^8y_0^2-1260R^8x_0^6y_0^4-1260R^8x_0^4y_0^6-630R^8x_0^2y_0^8-126R^8y_0^{10}+128R^7r^7x_0^4+256R^7r^7x_0^2y_0^2+128R^7r^7y_0^4-448R^7r^5x_0^6-1344R^7r^5x_0^4y_0^2-1344R^7r^5x_0^2y_0^4-448R^7r^5y_0^6+1000R^7r^3x_0^8+4000R^7r^3x_0^6y_0^2+6000R^7r^3x_0^4y_0^4+4000R^7r^3x_0^2y_0^6+1000R^7r^3y_0^8-336R^7rx_0^{10}-1680R^7rx_0^8y_0^2-3360R^7rx_0^6y_0^4-3360R^7rx_0^4y_0^6-1680R^7rx_0^2y_0^8-336R^7ry_0^{10}+960R^6r^4x_0^8+3840R^6r^4x_0^6y_0^2+5760R^6r^4x_0^4y_0^4+3840R^6r^4x_0^2y_0^6+960R^6r^4y_0^8+84R^6x_0^{12}+504R^6x_0^{10}y_0^2+1260R^6x_0^8y_0^4+1680R^6x_0^6y_0^6+1260R^6x_0^4y_0^8+504R^6x_0^2y_0^{10}+84R^6y_0^{12}-384R^5r^7x_0^6-1152R^5r^7x_0^4y_0^2-1152R^5r^7x_0^2y_0^4-384R^5r^7y_0^6+832R^5r^5x_0^8+3328R^5r^5x_0^6y_0^2+4992R^5r^5x_0^4y_0^4+3328R^5r^5x_0^2y_0^6+832R^5r^5y_0^8-792R^5r^3x_0^{10}-3960R^5r^3x_0^8y_0^2-7920R^5r^3x_0^6y_0^4-7920R^5r^3x_0^4y_0^6-3960R^5r^3x_0^2y_0^8-792R^5r^3y_0^{10}+168R^5rx_0^{12}+1008R^5rx_0^{10}y_0^2+2520R^5rx_0^8y_0^4+3360R^5rx_0^6y_0^6+2520R^5rx_0^4y_0^8+1008R^5rx_0^2y_0^{10}+168R^5ry_0^{12}+512R^4r^6x_0^8+2048R^4r^6x_0^6y_0^2+3072R^4r^6x_0^4y_0^4+2048R^4r^6x_0^2y_0^6+512R^4r^6y_0^8-480R^4r^4x_0^{10}-2400R^4r^4x_0^8y_0^2-4800R^4r^4x_0^6y_0^4-4800R^4r^4x_0^4y_0^6-2400R^4r^4x_0^2y_0^8-480R^4r^4y_0^{10}-36R^4x_0^{14}-252R^4x_0^{12}y_0^2-756R^4x_0^{10}y_0^4-1260R^4x_0^8y_0^6-1260R^4x_0^6y_0^8-756R^4x_0^4y_0^{10}-252R^4x_0^2y_0^{12}-36R^4y_0^{14}-512R^3r^9x_0^6-1536R^3r^9x_0^4y_0^2-1536R^3r^9x_0^2y_0^4-512R^3r^9y_0^6+384R^3r^7x_0^8+1536R^3r^7x_0^6y_0^2+2304R^3r^7x_0^4y_0^4+1536R^3r^7x_0^2y_0^6+384R^3r^7y_0^8-608R^3r^5x_0^{10}-3040R^3r^5x_0^8y_0^2-6080R^3r^5x_0^6y_0^4-6080R^3r^5x_0^4y_0^6-3040R^3r^5x_0^2y_0^8-608R^3r^5y_0^{10}+328R^3r^3x_0^{12}+1968R^3r^3x_0^{10}y_0^2+4920R^3r^3x_0^8y_0^4+6560R^3r^3x_0^6y_0^6+4920R^3r^3x_0^4y_0^8+1968R^3r^3x_0^2y_0^{10}+328R^3r^3y_0^{12}-48R^3rx_0^{14}-336R^3rx_0^{12}y_0^2-1008R^3rx_0^{10}y_0^4-1680R^3rx_0^8y_0^6-1680R^3rx_0^6y_0^8-1008R^3rx_0^4y_0^{10}-336R^3rx_0^2y_0^{12}-48R^3ry_0^{14}+256R^2r^8x_0^8+1024R^2r^8x_0^6y_0^2+1536R^2r^8x_0^4y_0^4+1024R^2r^8x_0^2y_0^6+256R^2r^8y_0^8-256R^2r^6x_0^{10}-1280R^2r^6x_0^8y_0^2-2560R^2r^6x_0^6y_0^4-2560R^2r^6x_0^4y_0^6-1280R^2r^6x_0^2y_0^8-256R^2r^6y_0^{10}+96R^2r^4x_0^{12}+576R^2r^4x_0^{10}y_0^2+1440R^2r^4x_0^8y_0^4+1920R^2r^4x_0^6y_0^6+1440R^2r^4x_0^4y_0^8+576R^2r^4x_0^2y_0^{10}+96R^2r^4y_0^{12}+9R^2x_0^{16}+72R^2x_0^{14}y_0^2+252R^2x_0^{12}y_0^4+504R^2x_0^{10}y_0^6+630R^2x_0^8y_0^8+504R^2x_0^6y_0^{10}+252R^2x_0^4y_0^{12}+72R^2x_0^2y_0^{14}+9R^2y_0^{16}-128Rr^7x_0^{10}-640Rr^7x_0^8y_0^2-1280Rr^7x_0^6y_0^4-1280Rr^7x_0^4y_0^6-640Rr^7x_0^2y_0^8-128Rr^7y_0^{10}+160Rr^5x_0^{12}+960Rr^5x_0^{10}y_0^2+2400Rr^5x_0^8y_0^4+3200Rr^5x_0^6y_0^6+2400Rr^5x_0^4y_0^8+960Rr^5x_0^2y_0^{10}+160Rr^5y_0^{12}-56Rr^3x_0^{14}-392Rr^3x_0^{12}y_0^2-1176Rr^3x_0^{10}y_0^4-1960Rr^3x_0^8y_0^6-1960Rr^3x_0^6y_0^8-1176Rr^3x_0^4y_0^{10}-392Rr^3x_0^2y_0^{12}-56Rr^3y_0^{14}+6Rrx_0^{16}+48Rrx_0^{14}y_0^2+168Rrx_0^{12}y_0^4+336Rrx_0^{10}y_0^6+420Rrx_0^8y_0^8+336Rrx_0^6y_0^{10}+168Rrx_0^4y_0^{12}+48Rrx_0^2y_0^{14}+6Rry_0^{16}-x_0^{18}-9x_0^{16}y_0^2-36x_0^{14}y_0^4-84x_0^{12}y_0^6-126x_0^{10}y_0^8-126x_0^8y_0^{10}-84x_0^6y_0^{12}-36x_0^4y_0^{14}-9x_0^2y_0^{16}-y_0^{18}=0

复制代码

\(n=10\)时

1. 5R^24-20R^22r^2-60R^22x_0^2-60R^22y_0^2+16R^{20}r^4+180R^{20}r^2x_0^2+180R^{20}r^2y_0^2+330R^{20}x_0^4+660R^{20}x_0^2y_0^2+330R^{20}y_0^4-304R^{18}r^4x_0^2-304R^{18}r^4y_0^2-700R^{18}r^2x_0^4-1400R^{18}r^2x_0^2y_0^2-700R^{18}r^2y_0^4-1100R^{18}x_0^6-3300R^{18}x_0^4y_0^2-3300R^{18}x_0^2y_0^4-1100R^{18}y_0^6+1152R^{16}r^6x_0^2+1152R^{16}r^6y_0^2+1872R^{16}r^4x_0^4+3744R^{16}r^4x_0^2y_0^2+1872R^{16}r^4y_0^4+1500R^{16}r^2x_0^6+4500R^{16}r^2x_0^4y_0^2+4500R^{16}r^2x_0^2y_0^4+1500R^{16}r^2y_0^6+2475R^{16}x_0^8+9900R^{16}x_0^6y_0^2+14850R^{16}x_0^4y_0^4+9900R^{16}x_0^2y_0^6+2475R^{16}y_0^8-1792R^{14}r^8x_0^2-1792R^{14}r^8y_0^2-5760R^{14}r^6x_0^4-11520R^{14}r^6x_0^2y_0^2-5760R^{14}r^6y_0^4-5952R^{14}r^4x_0^6-17856R^{14}r^4x_0^4y_0^2-17856R^{14}r^4x_0^2y_0^4-5952R^{14}r^4y_0^6-1800R^{14}r^2x_0^8-7200R^{14}r^2x_0^6y_0^2-10800R^{14}r^2x_0^4y_0^4-7200R^{14}r^2x_0^2y_0^6-1800R^{14}r^2y_0^8-3960R^{14}x_0^{10}-19800R^{14}x_0^8y_0^2-39600R^{14}x_0^6y_0^4-39600R^{14}x_0^4y_0^6-19800R^{14}x_0^2y_0^8-3960R^{14}y_0^{10}+1024R^{12}r^{10}x_0^2+1024R^{12}r^{10}y_0^2+3328R^{12}r^8x_0^4+6656R^{12}r^8x_0^2y_0^2+3328R^{12}r^8y_0^4+10368R^{12}r^6x_0^6+31104R^{12}r^6x_0^4y_0^2+31104R^{12}r^6x_0^2y_0^4+10368R^{12}r^6y_0^6+11424R^{12}r^4x_0^8+45696R^{12}r^4x_0^6y_0^2+68544R^{12}r^4x_0^4y_0^4+45696R^{12}r^4x_0^2y_0^6+11424R^{12}r^4y_0^8+840R^{12}r^2x_0^{10}+4200R^{12}r^2x_0^8y_0^2+8400R^{12}r^2x_0^6y_0^4+8400R^{12}r^2x_0^4y_0^6+4200R^{12}r^2x_0^2y_0^8+840R^{12}r^2y_0^{10}+4620R^{12}x_0^{12}+27720R^{12}x_0^{10}y_0^2+69300R^{12}x_0^8y_0^4+92400R^{12}x_0^6y_0^6+69300R^{12}x_0^4y_0^8+27720R^{12}x_0^2y_0^{10}+4620R^{12}y_0^{12}+2816R^{10}r^8x_0^6+8448R^{10}r^8x_0^4y_0^2+8448R^{10}r^8x_0^2y_0^4+2816R^{10}r^8y_0^6-5760R^{10}r^6x_0^8-23040R^{10}r^6x_0^6y_0^2-34560R^{10}r^6x_0^4y_0^4-23040R^{10}r^6x_0^2y_0^6-5760R^{10}r^6y_0^8-14112R^{10}r^4x_0^{10}-70560R^{10}r^4x_0^8y_0^2-141120R^{10}r^4x_0^6y_0^4-141120R^{10}r^4x_0^4y_0^6-70560R^{10}r^4x_0^2y_0^8-14112R^{10}r^4y_0^{10}+840R^{10}r^2x_0^{12}+5040R^{10}r^2x_0^{10}y_0^2+12600R^{10}r^2x_0^8y_0^4+16800R^{10}r^2x_0^6y_0^6+12600R^{10}r^2x_0^4y_0^8+5040R^{10}r^2x_0^2y_0^{10}+840R^{10}r^2y_0^{12}-3960R^{10}x_0^{14}-27720R^{10}x_0^{12}y_0^2-83160R^{10}x_0^{10}y_0^4-138600R^{10}x_0^8y_0^6-138600R^{10}x_0^6y_0^8-83160R^{10}x_0^4y_0^{10}-27720R^{10}x_0^2y_0^{12}-3960R^{10}y_0^{14}-1024R^8r^{10}x_0^6-3072R^8r^{10}x_0^4y_0^2-3072R^8r^{10}x_0^2y_0^4-1024R^8r^{10}y_0^6-8704R^8r^8x_0^8-34816R^8r^8x_0^6y_0^2-52224R^8r^8x_0^4y_0^4-34816R^8r^8x_0^2y_0^6-8704R^8r^8y_0^8-5760R^8r^6x_0^{10}-28800R^8r^6x_0^8y_0^2-57600R^8r^6x_0^6y_0^4-57600R^8r^6x_0^4y_0^6-28800R^8r^6x_0^2y_0^8-5760R^8r^6y_0^{10}+11424R^8r^4x_0^{12}+68544R^8r^4x_0^{10}y_0^2+171360R^8r^4x_0^8y_0^4+228480R^8r^4x_0^6y_0^6+171360R^8r^4x_0^4y_0^8+68544R^8r^4x_0^2y_0^{10}+11424R^8r^4y_0^{12}-1800R^8r^2x_0^{14}-12600R^8r^2x_0^{12}y_0^2-37800R^8r^2x_0^{10}y_0^4-63000R^8r^2x_0^8y_0^6-63000R^8r^2x_0^6y_0^8-37800R^8r^2x_0^4y_0^{10}-12600R^8r^2x_0^2y_0^{12}-1800R^8r^2y_0^{14}+2475R^8x_0^{16}+19800R^8x_0^{14}y_0^2+69300R^8x_0^{12}y_0^4+138600R^8x_0^{10}y_0^6+173250R^8x_0^8y_0^8+138600R^8x_0^6y_0^{10}+69300R^8x_0^4y_0^{12}+19800R^8x_0^2y_0^{14}+2475R^8y_0^{16}+4096R^6r^{12}x_0^6+12288R^6r^{12}x_0^4y_0^2+12288R^6r^{12}x_0^2y_0^4+4096R^6r^{12}y_0^6-1024R^6r^{10}x_0^8-4096R^6r^{10}x_0^6y_0^2-6144R^6r^{10}x_0^4y_0^4-4096R^6r^{10}x_0^2y_0^6-1024R^6r^{10}y_0^8+2816R^6r^8x_0^{10}+14080R^6r^8x_0^8y_0^2+28160R^6r^8x_0^6y_0^4+28160R^6r^8x_0^4y_0^6+14080R^6r^8x_0^2y_0^8+2816R^6r^8y_0^{10}+10368R^6r^6x_0^{12}+62208R^6r^6x_0^{10}y_0^2+155520R^6r^6x_0^8y_0^4+207360R^6r^6x_0^6y_0^6+155520R^6r^6x_0^4y_0^8+62208R^6r^6x_0^2y_0^{10}+10368R^6r^6y_0^{12}-5952R^6r^4x_0^{14}-41664R^6r^4x_0^{12}y_0^2-124992R^6r^4x_0^{10}y_0^4-208320R^6r^4x_0^8y_0^6-208320R^6r^4x_0^6y_0^8-124992R^6r^4x_0^4y_0^{10}-41664R^6r^4x_0^2y_0^{12}-5952R^6r^4y_0^{14}+1500R^6r^2x_0^{16}+12000R^6r^2x_0^{14}y_0^2+42000R^6r^2x_0^{12}y_0^4+84000R^6r^2x_0^{10}y_0^6+105000R^6r^2x_0^8y_0^8+84000R^6r^2x_0^6y_0^{10}+42000R^6r^2x_0^4y_0^{12}+12000R^6r^2x_0^2y_0^{14}+1500R^6r^2y_0^{16}-1100R^6x_0^{18}-9900R^6x_0^{16}y_0^2-39600R^6x_0^{14}y_0^4-92400R^6x_0^{12}y_0^6-138600R^6x_0^{10}y_0^8-138600R^6x_0^8y_0^{10}-92400R^6x_0^6y_0^{12}-39600R^6x_0^4y_0^{14}-9900R^6x_0^2y_0^{16}-1100R^6y_0^{18}+3328R^4r^8x_0^{12}+19968R^4r^8x_0^{10}y_0^2+49920R^4r^8x_0^8y_0^4+66560R^4r^8x_0^6y_0^6+49920R^4r^8x_0^4y_0^8+19968R^4r^8x_0^2y_0^{10}+3328R^4r^8y_0^{12}-5760R^4r^6x_0^{14}-40320R^4r^6x_0^{12}y_0^2-120960R^4r^6x_0^{10}y_0^4-201600R^4r^6x_0^8y_0^6-201600R^4r^6x_0^6y_0^8-120960R^4r^6x_0^4y_0^{10}-40320R^4r^6x_0^2y_0^{12}-5760R^4r^6y_0^{14}+1872R^4r^4x_0^{16}+14976R^4r^4x_0^{14}y_0^2+52416R^4r^4x_0^{12}y_0^4+104832R^4r^4x_0^{10}y_0^6+131040R^4r^4x_0^8y_0^8+104832R^4r^4x_0^6y_0^{10}+52416R^4r^4x_0^4y_0^{12}+14976R^4r^4x_0^2y_0^{14}+1872R^4r^4y_0^{16}-700R^4r^2x_0^{18}-6300R^4r^2x_0^{16}y_0^2-25200R^4r^2x_0^{14}y_0^4-58800R^4r^2x_0^{12}y_0^6-88200R^4r^2x_0^{10}y_0^8-88200R^4r^2x_0^8y_0^{10}-58800R^4r^2x_0^6y_0^{12}-25200R^4r^2x_0^4y_0^{14}-6300R^4r^2x_0^2y_0^{16}-700R^4r^2y_0^{18}+330R^4x_0^{20}+3300R^4x_0^{18}y_0^2+14850R^4x_0^{16}y_0^4+39600R^4x_0^{14}y_0^6+69300R^4x_0^{12}y_0^8+83160R^4x_0^{10}y_0^{10}+69300R^4x_0^8y_0^{12}+39600R^4x_0^6y_0^{14}+14850R^4x_0^4y_0^{16}+3300R^4x_0^2y_0^{18}+330R^4y_0^{20}+1024R^2r^{10}x_0^{12}+6144R^2r^{10}x_0^{10}y_0^2+15360R^2r^{10}x_0^8y_0^4+20480R^2r^{10}x_0^6y_0^6+15360R^2r^{10}x_0^4y_0^8+6144R^2r^{10}x_0^2y_0^{10}+1024R^2r^{10}y_0^{12}-1792R^2r^8x_0^{14}-12544R^2r^8x_0^{12}y_0^2-37632R^2r^8x_0^{10}y_0^4-62720R^2r^8x_0^8y_0^6-62720R^2r^8x_0^6y_0^8-37632R^2r^8x_0^4y_0^{10}-12544R^2r^8x_0^2y_0^{12}-1792R^2r^8y_0^{14}+1152R^2r^6x_0^{16}+9216R^2r^6x_0^{14}y_0^2+32256R^2r^6x_0^{12}y_0^4+64512R^2r^6x_0^{10}y_0^6+80640R^2r^6x_0^8y_0^8+64512R^2r^6x_0^6y_0^{10}+32256R^2r^6x_0^4y_0^{12}+9216R^2r^6x_0^2y_0^{14}+1152R^2r^6y_0^{16}-304R^2r^4x_0^{18}-2736R^2r^4x_0^{16}y_0^2-10944R^2r^4x_0^{14}y_0^4-25536R^2r^4x_0^{12}y_0^6-38304R^2r^4x_0^{10}y_0^8-38304R^2r^4x_0^8y_0^{10}-25536R^2r^4x_0^6y_0^{12}-10944R^2r^4x_0^4y_0^{14}-2736R^2r^4x_0^2y_0^{16}-304R^2r^4y_0^{18}+180R^2r^2x_0^{20}+1800R^2r^2x_0^{18}y_0^2+8100R^2r^2x_0^{16}y_0^4+21600R^2r^2x_0^{14}y_0^6+37800R^2r^2x_0^{12}y_0^8+45360R^2r^2x_0^{10}y_0^{10}+37800R^2r^2x_0^8y_0^{12}+21600R^2r^2x_0^6y_0^{14}+8100R^2r^2x_0^4y_0^{16}+1800R^2r^2x_0^2y_0^{18}+180R^2r^2y_0^{20}-60R^2x_0^22-660R^2x_0^{20}y_0^2-3300R^2x_0^{18}y_0^4-9900R^2x_0^{16}y_0^6-19800R^2x_0^{14}y_0^8-27720R^2x_0^{12}y_0^{10}-27720R^2x_0^{10}y_0^{12}-19800R^2x_0^8y_0^{14}-9900R^2x_0^6y_0^{16}-3300R^2x_0^4y_0^{18}-660R^2x_0^2y_0^{20}-60R^2y_0^22+16r^4x_0^{20}+160r^4x_0^{18}y_0^2+720r^4x_0^{16}y_0^4+1920r^4x_0^{14}y_0^6+3360r^4x_0^{12}y_0^8+4032r^4x_0^{10}y_0^{10}+3360r^4x_0^8y_0^{12}+1920r^4x_0^6y_0^{14}+720r^4x_0^4y_0^{16}+160r^4x_0^2y_0^{18}+16r^4y_0^{20}-20r^2x_0^22-220r^2x_0^{20}y_0^2-1100r^2x_0^{18}y_0^4-3300r^2x_0^{16}y_0^6-6600r^2x_0^{14}y_0^8-9240r^2x_0^{12}y_0^{10}-9240r^2x_0^{10}y_0^{12}-6600r^2x_0^8y_0^{14}-3300r^2x_0^6y_0^{16}-1100r^2x_0^4y_0^{18}-220r^2x_0^2y_0^{20}-20r^2y_0^22+5x_0^24+60x_0^22y_0^2+330x_0^{20}y_0^4+1100x_0^{18}y_0^6+2475x_0^{16}y_0^8+3960x_0^{14}y_0^{10}+4620x_0^{12}y_0^{12}+3960x_0^{10}y_0^{14}+2475x_0^8y_0^{16}+1100x_0^6y_0^{18}+330x_0^4y_0^{20}+60x_0^2y_0^22+5y_0^24=0

复制代码

若进一步要求\( x_0=0, y_0=0\),结果进一步简化为

\(n=3\)时

\(R-2r=0\)


\(n=4\)时

\(R^2-2r^2=0\)


\(n=5\)时

\(R^2+2Rr-4r^2=0\)


\(n=6\)时

\(3R^2-4r^2=0\)


\(n=7\)时

\(R^3-4R^2r-4Rr^2+8r^3=0\)


\(n=8\)时

\(R^4-8R^2r^2+8r^4=0\)


\(n=9\)时

\(R^3+6R^2r-8r^3=0\)


\(n=10\)时

\(5R^4-20R^2r^2+16r^4=0\)

数学星空

若记楼上的结论\(x_0^2+y_0^2=d^2\),即可以得到双圆内接外切N边形的圆心心距公式：

当\(n=3\)时

\(R^2-2Rr-d^2＝0\)

当\(n=4\)时

\(d^4+(-2R^2-2r^2)d^2+R^4-2r^2R^2＝0\)

当\(n=5\)时

\(-d^6+(3R^2+2Rr)d^4+(-3R^4-4R^3r+4R^2r^2-8Rr^3)d^2+R^6+2R^5r-4R^4r^2＝0\)

当\(n=6\)时

\(3d^8+(-12R^2-4r^2)d^6+(18R^4+4R^2r^2)d^4+(-12R^6+4R^4r^2-16R^2r^4)d^2+3R^8-4R^6r^2＝0\)

当\(n=7\)时

\(d^{12}+(-6R^2+4Rr)d^{10}+(15R^4-20R^3r-4R^2r^2-24Rr^3)d^8+(-20R^6+40R^5r+16R^4r^2+64R^3r^3-16R^2r^4+32Rr^5)d^6+(15R^8-40R^7r-24R^6r^2-48R^5r^3+32R^4r^4)d^4+(-6R^{10}+20R^9r+16R^8r^2-16R^6r^4-32R^5r^5+64R^4r^6)d^2+R^{12}-4R^{11}r-4R^{10}r^2+8R^9r^3＝0\)

当\(n=8\)时

\(d^{16}+(-8R^2-8r^2)d^{14}+(28R^4+40R^2r^2+8r^4)d^{12}+(-56R^6-72R^4r^2+48R^2r^4)d^{10}+(70R^8+40R^6r^2-264R^4r^4-128R^2r^6)d^8+(-56R^{10}+40R^8r^2+416R^6r^4+128R^4r^6+128R^2r^8)d^6+(28R^{12}-72R^{10}r^2-264R^8r^4+128R^6r^6)d^4+(-8R^{14}+40R^{12}r^2+48R^{10}r^4-128R^8r^6+128R^6r^8)d^2+R^{16}-8R^{14}r^2+8R^{12}r^4＝0\)

当\(n=9\)时

\(-d^{18}+(9R^2+6Rr)d^{16}+(-36R^4-48R^3r-56Rr^3)d^{14}+(84R^6+168R^5r+328R^3r^3+96R^2r^4+160Rr^5)d^{12}+(-126R^8-336R^7r-792R^5r^3-480R^4r^4-608R^3r^5-256R^2r^6-128Rr^7)d^{10}+(126R^{10}+420R^9r+1000R^7r^3+960R^6r^4+832R^5r^5+512R^4r^6+384R^3r^7+256R^2r^8)d^8+(-84R^{12}-336R^{11}r-680R^9r^3-960R^8r^4-448R^7r^5-384R^5r^7-512R^3r^9)d^6+(36R^{14}+168R^{13}r+216R^{11}r^3+480R^{10}r^4+32R^9r^5-512R^8r^6+128R^7r^7)d^4+(-9R^{16}-48R^{16}r-8R^{13}r^3-96R^{12}r^4+32R^{11}r^5+256R^{10}r^6-256R^8r^8)d^2+R^{18}+6R^{17}r-8R^{15}r^3＝0\)

当\(n=10\)时

\(5d^{24}+(-60R^2-20r^2)d^{22}+(330R^4+180R^2r^2+16r^4)d^{20}+(-1100R^6-700R^4r^2-304R^2r^4)d^{18}+(2475R^8+1500R^6r^2+1872R^4r^4+1152R^2r^6)d^{16}+(-3960R^{10}-1800R^8r^2-5952R^6r^4-5760R^4r^6-1792R^2r^8)d^{14}+(4620R^{12}+840R^{10}r^2+11424R^8r^4+10368R^6r^6+3328R^4r^8+1024R^2r^{10})d^{12}+(-3960R^{14}+840R^{12}r^2-14112R^{10}r^4-5760R^8r^6+2816R^6r^8)d^{10}+(2475R^{16}-1800R^{14}r^2+11424R^{12}r^4-5760R^{10}r^6-8704R^8r^8-1024R^6r^{10})d^8+(-1100R^{18}+1500R^{16}r^2-5952R^{14}r^4+10368R^{12}r^6+2816R^{10}r^8-1024R^8r^{10}+4096R^6r^{12})d^6+(330R^{20}-700R^{18}r^2+1872R^{16}r^4-5760R^{14}r^6+3328R^{12}r^8)d^4+(-60R^{22}+180R^{20}r^2-304R^{18}r^4+1152R^{16}r^6-1792R^{14}r^8+1024R^{12}r^{10})d^2+5R^{24}-20R^{22}r^2+16R^{20}r^4＝0\)

zuijianqiugen

数学星空 发表于 2014-5-1 13:37
若记楼上的结论\(x_0^2+y_0^2=d^2\),即可以得到双圆内接外切N边形的圆心心距公式：

当\(n=3\)时

你有计算软件，能否做一些进一步的简化工作？参见《双圆多边形的心距公式  》：
http://zuijianqiugen.blog.163.co ... 240622011615127734/

数学星空

根据楼上的简化思路，我将13#的结果按照楼上的结论重写一下：

当\(n=3\)时

\(d=\sqrt{R^2-2Rr}\)

对称写成：\(\frac{1}{R+d}+\frac{1}{R-d}=\frac{1}{r}\)

当\(n=4\)时

\(d=\sqrt{R^2-2tRr}\)    且 \(\frac{1-t^2}{t}=\frac{r}{R}\)

对称写成：\(\frac{1}{(R+d)^2}+\frac{1}{(R-d)^2}=\frac{1}{r^2}\)

当\(n=5\)时

\(d=\sqrt{R^2-2tRr}\)    且 \(\frac{(1+t)(1-t^2)}{t}=\frac{2r}{R}\)

当\(n=6\)时

\(d=\sqrt{R^2-2tRr}\)    且 \(\frac{(1-t^2)(1+3t^2)}{(1+t^2)t}=\frac{2r}{R}\)

当\(n=7\)时

\(d=\sqrt{R^2-2tRr}\)    且 \(\frac{(1-t^2)(t-1)^2}{(1-3t+\sqrt{(t+1)(4t^2-3t+1)})t}=\frac{r}{R}\)

当\(n=8\)时

\(d=\sqrt{R^2-2tRr}\)    且\(x=\frac{R}{2r}\) ,  \(2(t-1)^4(t+1)^4x^3+4t(t-1)^3(t+1)^3x^2+t^2(t-1)(t+1)(t^2-3)x-t^3=0\)

zuijianqiugen

数学星空 发表于 2014-5-1 21:47
根据楼上的简化思路，我将13#的结果按照楼上的结论重写一下：

当\(n=3\)时

当n=7时，d的最高次数为12，应该化成t的六次方程，而你提供的是t的八次方程。不知是软件问题，还是笔误？

数学星空

对于15#的简化结果：

我们可以换一种思路来解释：

当\(n=5\)时

由\(d=0\)得到\(t=\frac{R}{2r}=\frac{1}{2cos(\frac{\pi}{5})}\),

则有 \(\frac{1}{t}＝\frac{(1+t)(1-t^2)}{t}\)

我们移项并 分母有理化后取分母分解因式为：\(t(t^2+t-1)＝0\)

又因为 以\(\frac{1}{2cos(\frac{\pi}{5})}\)为根的最小有理方程为\(t^2+t-1=0\)




当\(n=6\)时

由\(d=0\)得到\(t=\frac{R}{2r}=\frac{1}{2cos(\frac{\pi}{6})}\),

得到：\(\frac{1}{t}=\frac{(1-t^2)(1+3t^2)}{t(1+t^2)}\)

移项取分母并分解因式：\(t^2(3t^2-1)=0\)

显然以\(\frac{1}{2cos(\frac{\pi}{6})}\)为根的最小有理方程为\(3t^2-1=0\)




当\(n=7\)时

由\(d=0\)得到\(t=\frac{R}{2r}=\frac{1}{2cos(\frac{\pi}{7})}\),

得到：\(\frac{1}{t}=\frac{2(1-t^2)(1-t)^2}{t(1-3t+\sqrt{(1+t)(4t^2-3t+1)}}\)

移项取分母并将其有理化分解因式：\(4t^3(t^3-2t^2-t+1)(-1+t)^2＝0\)

又因为 以\(\frac{1}{2cos(\frac{\pi}{7})}\)为根的最小有理方程为\(t^3-2t^2-t+1＝0\)

zuijianqiugen

数学星空 发表于 2014-5-1 23:35
对于15#的简化结果：

我们可以换一种思路来解释：

我说的意思：当n=7时，你化简的关于t的方程是错误的。不知是软件问题，还是笔误？

zuijianqiugen

数学星空 发表于 2014-5-1 23:35
对于15#的简化结果：

我们可以换一种思路来解释：

当n=7时，d的最高次数为12，而d²=R²-2tRr，消去d后应该化成t的六次方程。

数学星空

当\(n=7\)时

消元结果为：\(64x^4r^{12}( t^6x^2-2t^5x^2-t^4x^2-6t^4x+4t^3x^2+2t^3x-t^2x^2-4t^3+6t^2x-2tx^2-2tx+x^2)＝0\)，且 ,\(x=\frac{R}{r}\)

求解\(x\)关于\(t\)的方程得到：\(\D\frac{1}{x}=\frac{(1-t^2)(t-1)^2}{\Big(1-3t+\sqrt{(t+1)(4t^2-3t+1)}\Big)t}=\frac{r}{R}\)

注：

\(d^{12}+(-6R^2+4Rr)d^{10}+(15R^4-20R^3r-4R^2r^2-24Rr^3)d^8+(-20R^6+40R^5r+16R^4r^2+64R^3r^3-16R^2r^4+32Rr^5)d^6+(15R^8-40R^7r-24R^6r^2-48R^5r^3+32R^4r^4)d^4+(-6R^{10}+20R^9r+16R^8r^2-16R^6r^4-32R^5r^5+64R^4r^6)d^2+R^{12}-4R^{11}r-4R^{10}r^2+8R^9r^3＝0\)

与\(d^2-R^2+2trR=0\)的消元结果为：\(64R^4r^6( t^6x^2-2t^5x^2-t^4x^2-6t^4x+4t^3x^2+2t^3x-t^2x^2-4t^3+6t^2x-2tx^2-2tx+x^2)=0\)

运算结果是正确的。