圆内接五边形的面积公式

数学星空

更进一步简化：
\(s_5^3-s_5^2(2s_1s_3-s_2^2+4s_4)R^2+s_5(2s_1^3s_5-2s_1^2s_2s_4+s_1^2s_3^2-8s_1s_2s_5+8s_1s_3s_4-2s_2s_3^2+32s_3s_5)R^4+(-2s_1^4s_3s_5+s_1^4s_4^2+8s_1^3s_4s_5+4s_1^2s_2s_3s_5-2s_1^2s_3^2s_4-16s_1^2s_5^2-32s_1s_3^2s_5+16s_2^2s_3s_5+s_3^4-32s_2s_5^2-64s_3s_4s_5)R^6+(s_1^6s_5+6s_1^4s_2s_5-4s_1^4s_3s_4+32s_1^3s_3s_5-32s_1^3s_4^2-32s_1^2s_2^2s_5+16s_1^2s_2s_3s_4+4s_1^2s_3^3-32s_1^2s_4s_5+32s_1s_3^2s_4-32s_2^3s_5-16s_2s_3^3+256s_1s_5^2+128s_2s_4s_5+224s_3^2s_5)R^8+(-2s_1^6s_4-64s_1^5s_5+16s_1^4s_2s_4+6s_1^4s_3^2+128s_1^3s_2s_5+64s_1^3s_3s_4-32s_1^2s_2^2s_4-48s_1^2s_2s_3^2-576s_1^2s_3s_5+384s_1^2s_4^2+512s_1s_2^2s_5-256s_1s_2s_3s_4+96s_2^2s_3^2-512s_1s_4s_5-768s_2s_3s_5-128s_3^2s_4-768s_5^2)R^{10}+(4s_1^6s_3+32s_1^5s_4-48s_1^4s_2s_3+736s_1^4s_5-256s_1^3s_2s_4+192s_1^2s_2^2s_3-2816s_1^2s_2s_5-256s_1^2s_3s_4+512s_1s_2^2s_4-256s_2^3s_3+6144s_1s_3s_5-2048s_1s_4^2-512s_2^2s_5+1024s_2s_3s_4+2048s_4s_5)R^{12}+(s_1^8-16s_1^6s_2+96s_1^4s_2^2-128s_1^4s_4-256s_1^2s_2^3-2048s_1^3s_5+1024s_1^2s_2s_4+256s_2^4+8192s_1s_2s_5-2048s_2^2s_4-16384s_3s_5+4096s_4^2)R^{14}=0\)

且：
\(s_1 = a^2+b^2+c^2+d^2+e^2\)
\(s_2 = a^2(b^2+c^2+d^2+e^2)+b^2(c^2+d^2+e^2)+c^2(d^2+e^2)+d^2e^2\)
\(s_3 = a^2b^2(c^2+d^2+e^2)+a^2c^2(d^2+e^2)+a^2d^2e^2+b^2c^2(d^2+e^2)+b^2d^2e^2+c^2d^2e^2\)
\(s_4 = a^2b^2c^2(d^2+e^2)+a^2b^2d^2e^2+a^2c^2d^2e^2+b^2c^2d^2e^2\)
\(s_5 = a^2b^2c^2d^2e^2\)

云梦

虽然上面出现了一点问题，右边的公式用四边形的面积替代就可以了，但不影响思路。
只要解出下列方程，就ok!
下面其实是一个三角形与一个四边形面积之和与五边形面积的差。构成一个一元方程。

数学星空

对于$sqrt(a^2*(4*R^2-a^2))+sqrt(b^2*(4*R^2-b^2))+sqrt(c^2*(4*R^2-c^2))+sqrt(d^2*(4*R^2-d^2))+sqrt(e^2*(4*R^2-e^2))=4*s$转化为代数方程问题
今天终于得到
(s1^8-16*s1^6*s2+96*s1^4*s2^2-128*s1^4*s4-256*s1^2*s2^3-2048*s1^3*s5+1024*s1^2*s2*s4+256*s2^4+8192*s1*s2*s5-2048*s2^2*s4-16384*s3*s5+4096*s4^2)^2-(256*(s1^15-28*s1^13*s2+16*s1^12*s3+336*s1^11*s2^2+64*s1^11*s4-384*s1^10*s2*s3-2240*s1^9*s2^3+5376*s1^10*s5-1280*s1^9*s2*s4+3840*s1^8*s2^2*s3+8960*s1^7*s2^4-80896*s1^8*s2*s5-1024*s1^8*s3*s4+10240*s1^7*s2^2*s4-20480*s1^6*s2^3*s3-21504*s1^5*s2^5+147456*s1^7*s3*s5-20480*s1^7*s4^2+434176*s1^6*s2^2*s5+16384*s1^6*s2*s3*s4-40960*s1^5*s2^3*s4+61440*s1^4*s2^4*s3+28672*s1^3*s2^6+229376*s1^6*s4*s5-1769472*s1^5*s2*s3*s5+245760*s1^5*s2*s4^2-884736*s1^4*s2^3*s5-98304*s1^4*s2^2*s3*s4+81920*s1^3*s2^4*s4-98304*s1^2*s2^5*s3-16384*s1*s2^7+2097152*s1^5*s5^2-2490368*s1^4*s2*s4*s5+786432*s1^4*s3^2*s5-65536*s1^4*s3*s4^2+7077888*s1^3*s2^2*s3*s5-983040*s1^3*s2^2*s4^2+65536*s1^2*s2^4*s5+262144*s1^2*s2^3*s3*s4-65536*s1*s2^5*s4+65536*s2^6*s3-12582912*s1^3*s2*s5^2-1048576*s1^3*s3*s4*s5+786432*s1^3*s4^3+8912896*s1^2*s2^2*s4*s5-6291456*s1^2*s2*s3^2*s5+524288*s1^2*s2*s3*s4^2-9437184*s1*s2^3*s3*s5+1310720*s1*s2^3*s4^2+1310720*s2^5*s5-262144*s2^4*s3*s4+12582912*s1^2*s3*s5^2-3145728*s1^2*s4^2*s5+16777216*s1*s2^2*s5^2+4194304*s1*s2*s3*s4*s5-3145728*s1*s2*s4^3-10485760*s2^3*s4*s5+12582912*s2^2*s3^2*s5-1048576*s2^2*s3*s4^2-33554432*s1*s4*s5^2-16777216*s2*s3*s5^2+20971520*s2*s4^2*s5-16777216*s3^2*s4*s5+4194304*s3*s4^3+67108864*s5^3))*s^2+(2048*(15*s1^14-364*s1^12*s2+384*s1^11*s3+3696*s1^10*s2^2+2496*s1^10*s4-7680*s1^9*s2*s3-20160*s1^8*s2^3+66048*s1^9*s5-41216*s1^8*s2*s4+3072*s1^8*s3^2+61440*s1^7*s2^2*s3+62720*s1^6*s2^4-942080*s1^7*s2*s5+49152*s1^7*s3*s4+260096*s1^6*s2^2*s4-49152*s1^6*s2*s3^2-245760*s1^5*s2^3*s3-107520*s1^4*s2^5+344064*s1^6*s3*s5+282624*s1^6*s4^2+4964352*s1^5*s2^2*s5-589824*s1^5*s2*s3*s4-761856*s1^4*s2^3*s4+294912*s1^4*s2^2*s3^2+491520*s1^3*s2^4*s3+86016*s1^2*s2^6-1245184*s1^5*s4*s5-3997696*s1^4*s2*s3*s5-2048000*s1^4*s2*s4^2+131072*s1^4*s3^2*s4-11403264*s1^3*s2^3*s5+2359296*s1^3*s2^2*s3*s4+966656*s1^2*s2^4*s4-786432*s1^2*s2^3*s3^2-393216*s1*s2^5*s3-16384*s2^7+19398656*s1^4*s5^2+8912896*s1^3*s2*s4*s5-4194304*s1^3*s3^2*s5+3670016*s1^3*s3*s4^2+15466496*s1^2*s2^2*s3*s5+2818048*s1^2*s2^2*s4^2-1048576*s1^2*s2*s3^2*s4+9568256*s1*s2^4*s5-3145728*s1*s2^3*s3*s4-327680*s2^5*s4+786432*s2^4*s3^2-100663296*s1^2*s2*s5^2-9437184*s1^2*s3*s4*s5+1310720*s1^2*s4^3-15728640*s1*s2^2*s4*s5+16777216*s1*s2*s3^2*s5-14680064*s1*s2*s3*s4^2-19922944*s2^3*s3*s5+3407872*s2^3*s4^2+2097152*s2^2*s3^2*s4+25165824*s1*s3*s5^2+44040192*s1*s4^2*s5+125829120*s2^2*s5^2+12582912*s2*s3*s4*s5-7340032*s2*s4^3-16777216*s3^3*s5+12582912*s3^2*s4^2-201326592*s4*s5^2))*s^4-(65536*(35*s1^13-728*s1^11*s2+1056*s1^10*s3+6160*s1^9*s2^2+5440*s1^9*s4-17280*s1^8*s2*s3-26880*s1^7*s2^3+47872*s1^8*s5-72704*s1^7*s2*s4+12288*s1^7*s3^2+107520*s1^6*s2^2*s3+62720*s1^5*s2^4-765952*s1^6*s2*s5+135168*s1^6*s3*s4+350208*s1^5*s2^2*s4-147456*s1^5*s2*s3^2-307200*s1^4*s2^3*s3-71680*s1^3*s2^5-344064*s1^5*s3*s5+315392*s1^5*s4^2+4136960*s1^4*s2^2*s5-1196032*s1^4*s2*s3*s4+65536*s1^4*s3^3-704512*s1^3*s2^3*s4+589824*s1^3*s2^2*s3^2+368640*s1^2*s2^4*s3+28672*s1*s2^6+1671168*s1^4*s4*s5-131072*s1^3*s2*s3*s5-2850816*s1^3*s2*s4^2+1310720*s1^3*s3^2*s4-8585216*s1^2*s2^3*s5+3080192*s1^2*s2^2*s3*s4-524288*s1^2*s2*s3^3+475136*s1*s2^4*s4-786432*s1*s2^3*s3^2-98304*s2^5*s3+10485760*s1^3*s5^2-7077888*s1^2*s2*s4*s5-2621440*s1^2*s3^2*s5+1703936*s1^2*s3*s4^2+6029312*s1*s2^2*s3*s5+6356992*s1*s2^2*s4^2-5242880*s1*s2*s3^2*s4+4915200*s2^4*s5-1835008*s2^3*s3*s4+1048576*s2^2*s3^3-20971520*s1*s2*s5^2+11534336*s1*s3*s4*s5-1310720*s1*s4^3-2621440*s2^2*s4*s5-2097152*s2*s3^2*s5-7864320*s2*s3*s4^2+4194304*s3^3*s4-29360128*s3*s5^2+32505856*s4^2*s5))*s^6+(262144*(455*s1^12-8008*s1^10*s2+14080*s1^9*s3+55440*s1^8*s2^2+45120*s1^8*s4-184320*s1^7*s2*s3-188160*s1^6*s2^3-305152*s1^7*s5-480256*s1^6*s2*s4+172032*s1^6*s3^2+860160*s1^5*s2^2*s3+313600*s1^4*s2^4+483328*s1^5*s2*s5+950272*s1^5*s3*s4+1710080*s1^4*s2^2*s4-1474560*s1^4*s2*s3^2-1638400*s1^3*s2^3*s3-215040*s1^2*s2^5-3325952*s1^4*s3*s5-1273856*s1^4*s4^2+7372800*s1^3*s2^2*s5-6029312*s1^3*s2*s3*s4+1048576*s1^3*s3^3-2179072*s1^2*s2^3*s4+3538944*s1^2*s2^2*s3^2+983040*s1*s2^4*s3+28672*s2^6+15073280*s1^3*s4*s5-1179648*s1^2*s2*s3*s5+1671168*s1^2*s2*s4^2+5767168*s1^2*s3^2*s4-17694720*s1*s2^3*s5+8912896*s1*s2^2*s3*s4-4194304*s1*s2*s3^3+540672*s2^4*s4-1572864*s2^3*s3^2+95420416*s1^2*s5^2-74973184*s1*s2*s4*s5-16777216*s1*s3^2*s5-9437184*s1*s3*s4^2+39059456*s2^2*s3*s5+15269888*s2^2*s4^2-18874368*s2*s3^2*s4+4194304*s3^4-192937984*s2*s5^2+126877696*s3*s4*s5-4456448*s4^3))*s^8-(16777216*(273*s1^11-4004*s1^9*s2+7920*s1^8*s3+22176*s1^7*s2^2+10368*s1^7*s4-80640*s1^6*s2*s3-56448*s1^5*s2^3-347648*s1^6*s5-89600*s1^5*s2*s4+86016*s1^5*s3^2+268800*s1^4*s2^2*s3+62720*s1^3*s2^4+1914880*s1^4*s2*s5+92160*s1^4*s3*s4+235520*s1^3*s2^2*s4-491520*s1^3*s2*s3^2-307200*s1^2*s2^3*s3-21504*s1*s2^5-1720320*s1^3*s3*s5-798720*s1^3*s4^2-1843200*s1^2*s2^2*s5-540672*s1^2*s2*s3*s4+393216*s1^2*s3^3-172032*s1*s2^3*s4+589824*s1*s2^2*s3^2+61440*s2^4*s3+688128*s1^2*s4*s5+851968*s1*s2*s3*s5+3915776*s1*s2*s4^2-786432*s1*s3^2*s4-491520*s2^3*s5+425984*s2^2*s3*s4-524288*s2*s3^3+12582912*s1*s5^2-5111808*s2*s4*s5+3932160*s3^2*s5-2162688*s3*s4^2))*s^10+(134217728*(1001*s1^10-12012*s1^8*s2+25344*s1^7*s3+51744*s1^6*s2^2-8064*s1^6*s4-193536*s1^5*s2*s3-94080*s1^4*s2^3-982016*s1^5*s5+17920*s1^4*s2*s4+215040*s1^4*s3^2+430080*s1^3*s2^2*s3+62720*s1^2*s2^4+5316608*s1^3*s2*s5-442368*s1^3*s3*s4+30720*s1^2*s2^2*s4-737280*s1^2*s2*s3^2-245760*s1*s2^3*s3-7168*s2^5-6078464*s1^2*s3*s5+151552*s1^2*s4^2-4374528*s1*s2^2*s5+458752*s1*s2*s3*s4+524288*s1*s3^3-24576*s2^3*s4+294912*s2^2*s3^2+2031616*s1*s4*s5+2424832*s2*s3*s5+3883008*s2*s4^2-1966080*s3^2*s4+16252928*s5^2))*s^12-(4294967296*(715*s1^9-6864*s1^7*s2+14784*s1^6*s3+22176*s1^5*s2^2-23424*s1^5*s4-80640*s1^4*s2*s3-26880*s1^3*s2^3-313856*s1^4*s5+74752*s1^3*s2*s4+86016*s1^3*s3^2+107520*s1^2*s2^2*s3+8960*s1*s2^4+1740800*s1^2*s2*s5-323584*s1^2*s3*s4-38912*s1*s2^2*s4-147456*s1*s2*s3^2-20480*s2^3*s3-2539520*s1*s3*s5+700416*s1*s4^2-630784*s2^2*s5+147456*s2*s3*s4+65536*s3^3+1605632*s4*s5))*s^14+(8589934592*(6435*s1^8-48048*s1^6*s2+101376*s1^5*s3+110880*s1^4*s2^2-247680*s1^4*s4-368640*s1^3*s2*s3-80640*s1^2*s2^3-272384*s1^3*s5+498688*s1^2*s2*s4+344064*s1^2*s3^2+245760*s1*s2^2*s3+8960*s2^4+4562944*s1*s2*s5-1376256*s1*s3*s4-96256*s2^2*s4-196608*s2*s3^2-6045696*s3*s5+2371584*s4^2))*s^16-(1099511627776*(715*s1^7-4004*s1^5*s2+7920*s1^4*s3+6160*s1^3*s2^2-22720*s1^3*s4-17280*s1^2*s2*s3-2240*s1*s2^3+76032*s1^2*s5+23296*s1*s2*s4+12288*s1*s3^2+3840*s2^2*s3+74752*s2*s5-33792*s3*s4))*s^18+(8796093022208*(1001*s1^6-4004*s1^4*s2+7040*s1^3*s3+3696*s1^2*s2^2-20032*s1^2*s4-7680*s1*s2*s3-448*s2^3+88576*s1*s5+6912*s2*s4+3072*s3^2))*s^20-(281474976710656*(273*s1^5-728*s1^3*s2+1056*s1^2*s3+336*s1*s2^2-2496*s1*s4-384*s2*s3+7936*s5))*s^22+(1125899906842624*(455*s1^4-728*s1^2*s2+768*s1*s3+112*s2^2-1088*s4))*s^24-(72057594037927936*(35*s1^3-28*s1*s2+16*s3))*s^26+(576460752303423488*(15*s1^2-4*s2))*s^28-18446744073709551616*s1*s^30+18446744073709551616*s^32＝0

s1 = 4*R^2*s11-s11^2+2*s12,
s2 = 16*s12*R^4+(-4*s11*s12+12*s13)*R^2-2*s11*s13+s12^2+2*s14,
s3 = 64*s13*R^6+(-16*s11*s13+64*s14)*R^4+(-12*s11*s14+4*s12*s13+20*s15)*R^2-2*s11*s15+2*s12*s14-s13^2,
s4 = 256*s14*R^8+(-64*s11*s14+320*s15)*R^6+(-64*s11*s15+16*s12*s14)*R^4+(12*s12*s15-4*s13*s14)*R^2-2*s13*s15+s14^2,
s5 = 1024*R^10*s15-256*R^8*s11*s15+64*R^6*s12*s15-16*R^4*s13*s15+4*R^2*s14*s15-s15^2

s11 = a^2+b^2+c^2+d^2+e^2,
s12 = a^2*(b^2+c^2+d^2+e^2)+b^2*(c^2+d^2+e^2)+c^2*(d^2+e^2)+d^2*e^2,
s13 = a^2*b^2*(c^2+d^2+e^2)+a^2*c^2*(d^2+e^2)+a^2*d^2*e^2+b^2*c^2*(d^2+e^2)+b^2*d^2*e^2+c^2*d^2*e^2,
s14 = a^2*b^2*c^2*(d^2+e^2)+a^2*b^2*d^2*e^2+a^2*c^2*d^2*e^2+b^2*c^2*d^2*e^2,
s15 = a^2*b^2*c^2*d^2*e^2

云梦

密密麻麻！测试几个实例看看结果是否正确？

数学星空

可以取一个特例试试看：
取$a=b=c=d=e=1,R=1/(2*sin(pi/5))=sqrt(2)/sqrt(5-sqrt(5))$代入得到：

$s11=5, s12=10, s13=10, s14=5, s15=1$

$s1=-(5*(sqrt(5)+3))/(-5+sqrt(5))$

$s2=(20*(7+3*sqrt(5)))/(-5+sqrt(5))^2$

$s3=-(80*(4*sqrt(5)+9))/(-5+sqrt(5))^3$

$s4=(40*(47+21*sqrt(5)))/(-5+sqrt(5))^4$

$s5=-(16*(55*sqrt(5)+123))/(-5+sqrt(5))^5$

$-7190466112575-244034269322674176000000*s^20-102361190030597160960000000*sqrt(5)*s^26+7733612946055495680000000*sqrt(5)*s^24-3215674203525*sqrt(5)-18045096874129489920000000*s^24+92790105550356480000000*s^18-$

$1855708226458012876800000000*s^28-28317334663436697600000*s^16+7112084531543487283200000000*sqrt(5)*s^32-3754200649376045465600000000*sqrt(5)*s^30+829788231343013888000000000*sqrt(5)*s^28-$

$15903111004170695475200000000*s^32+287764912445184*sqrt(5)*s^2-458652774192644096000*sqrt(5)*s^12+54641156417388544000*sqrt(5)*s^10-11721617957836800*sqrt(5)*s^4+1241397006270726144000000*s^22-26210334560716800*s^4-$

$12652426126216396800000*sqrt(5)*s^16+2817741676221562880000*sqrt(5)*s^14-81344756440891392000000*sqrt(5)*s^20+40827646442156851200000*sqrt(5)*s^18-4715735923261440000*sqrt(5)*s^8+287254917645926400*sqrt(5)*s^6-$

$248279401254145228800000*sqrt(5)*s^22+6300980130244198400000*s^14+229430253516855705600000000*s^26-10544706180513792000*s^8+643461905767680*s^2+8394709705418604544000000000*s^30-$

$1025581667491774464000*s^12+642321523040256000*s^6+122181359281111040000*s^10$

$=(32768/78125*(-2207+987*sqrt(5)))*(-16*s^2+10*sqrt(5)+25)*(-80*s^2+18*sqrt(5)+45)^5*(-80*s^2+2*sqrt(5)+5)^10=0$

可以得到$s=1/4 sqrt(25+10 sqrt(5))=1.720477400 $(注：需要人为判断哪一个实根才是我们想要的根）

由于正五边形的面积（边长为1）：\(s=\frac{5\cot(\frac{\pi}{5})}{4}=\frac{5\sqrt{2}(\sqrt{5}+1)}{8\sqrt{5-\sqrt{5}}}=1.720477400\)

zeroieme

我推导的公式S只有14次，而且是偶次式。求验证

268435456*S^14+16777216*S^12*(7*F1^4-28*F1^2*F2+4*F2^2+48*F1*F3-48*F4)+1048576*S^10*(21*F1^8-168*F1^6*F2+360*F1^4*F2^2-96*F1^2*F2^3+288*F1^5*F3-1152*F1^3*F2*F3+192*F1*F2^2*F3+976*F1^2*F3^2-32*F2*F3^2-288*F1^4*F4+1120*F1^2*F2*F4-128*F2^2*F4-1920*F1*F3*F4+768*F4^2+32*F1^3*F5-64*F1*F2*F5+384*F3*F5)+65536*S^8*(35*F1^12-420*F1^10*F2+1740*F1^8*F2^2-2720*F1^6*F2^3+960*F1^4*F2^4+720*F1^9*F3-5760*F1^7*F2*F3+12480*F1^5*F2^2*F3-3840*F1^3*F2^3*F3+4880*F1^6*F3^2-19680*F1^4*F2*F3^2+4480*F1^2*F2^2*F3^2+10880*F1^3*F3^3-1280*F1*F2*F3^3+64*F3^4-720*F1^8*F4+5600*F1^6*F2*F4-11520*F1^4*F2^2*F4+2560*F1^2*F2^3*F4-9600*F1^5*F3*F4+37120*F1^3*F2*F3*F4-5120*F1*F2^2*F3*F4-31360*F1^2*F3^2*F4+1024*F2*F3^2*F4+3904*F1^4*F4^2-14336*F1^2*F2*F4^2+1024*F2^2*F4^2+24576*F1*F3*F4^2-4096*F4^3+160*F1^7*F5-960*F1^5*F2*F5+1280*F1^3*F2^2*F5+3072*F1^4*F3*F5-9728*F1^2*F2*F3*F5+1024*F2^2*F3*F5+12544*F1*F3^2*F5-1280*F1^3*F4*F5+2048*F1*F2*F4*F5-12288*F3*F4*F5-2048*F1^2*F5^2+4608*F2*F5^2)+4096*S^6*(35*F1^16-560*F1^14*F2+3440*F1^12*F2^2-9920*F1^10*F2^3+12800*F1^8*F2^4-5120*F1^6*F2^5+960*F1^13*F3-11520*F1^11*F2*F3+48000*F1^9*F2^2*F3-76800*F1^7*F2^3*F3+30720*F1^5*F2^4*F3+9760*F1^10*F3^2-78400*F1^8*F2*F3^2+174080*F1^6*F2^2*F3^2-66560*F1^4*F2^3*F3^2+43520*F1^7*F3^3-179200*F1^5*F2*F3^3+61440*F1^3*F2^2*F3^3+71936*F1^4*F3^4-21504*F1^2*F2*F3^4+2048*F1*F3^5-960*F1^12*F4+11200*F1^10*F2*F4-44800*F1^8*F2^2*F4+66560*F1^6*F2^3*F4-20480*F1^4*F2^4*F4-19200*F1^9*F3*F4+148480*F1^7*F2*F3*F4-307200*F1^5*F2^2*F3*F4+81920*F1^3*F2^3*F3*F4-125440*F1^6*F3^2*F4+485376*F1^4*F2*F3^2*F4-98304*F1^2*F2^2*F3^2*F4-266240*F1^3*F3^3*F4+32768*F1*F2*F3^3*F4-2048*F3^4*F4+7936*F1^8*F4^2-58368*F1^6*F2*F4^2+110592*F1^4*F2^2*F4^2-16384*F1^2*F2^3*F4^2+100352*F1^5*F3*F4^2-360448*F1^3*F2*F3*F4^2+32768*F1*F2^2*F3*F4^2+303104*F1^2*F3^2*F4^2-8192*F2*F3^2*F4^2-18432*F1^4*F4^3+57344*F1^2*F2*F4^3-98304*F1*F3*F4^3+320*F1^11*F5-3200*F1^9*F2*F5+10240*F1^7*F2^2*F5-10240*F1^5*F2^3*F5+8448*F1^8*F3*F5-57344*F1^6*F2*F3*F5+98304*F1^4*F2^2*F3*F5-16384*F1^2*F2^3*F3*F5+66560*F1^5*F3^2*F5-225280*F1^3*F2*F3^2*F5+32768*F1*F2^2*F3^2*F5+159744*F1^2*F3^3*F5-8192*F2*F3^3*F5-5120*F1^7*F4*F5+28672*F1^5*F2*F4*F5-32768*F1^3*F2^2*F4*F5-86016*F1^4*F3*F4*F5+237568*F1^2*F2*F3*F4*F5-16384*F2^2*F3*F4*F5-303104*F1*F3^2*F4*F5+16384*F1^3*F4^2*F5-16384*F1*F2*F4^2*F5+98304*F3*F4^2*F5-7936*F1^6*F5^2+49664*F1^4*F2*F5^2-79872*F1^2*F2^2*F5^2+16384*F2^3*F5^2-38912*F1^3*F3*F5^2+94208*F1*F2*F3*F5^2+30720*F3^2*F5^2+47104*F1^2*F4*F5^2-73728*F2*F4*F5^2-36864*F1*F5^3)+256*S^4*(21*F1^20-420*F1^18*F2+3420*F1^16*F2^2-14400*F1^14*F2^3+32640*F1^12*F2^4-36864*F1^10*F2^5+15360*F1^8*F2^6+720*F1^17*F3-11520*F1^15*F2*F3+71040*F1^13*F2^2*F3-207360*F1^11*F2^3*F3+276480*F1^9*F2^4*F3-122880*F1^7*F2^5*F3+9760*F1^14*F3^2-117440*F1^12*F2*F3^2+495360*F1^10*F2^2*F3^2-824320*F1^8*F2^3*F3^2+389120*F1^6*F2^4*F3^2+65280*F1^11*F3^3-529920*F1^9*F2*F3^3+1228800*F1^7*F2^2*F3^3-614400*F1^5*F2^3*F3^3+215424*F1^8*F3^4-924672*F1^6*F2*F3^4+497664*F1^4*F2^2*F3^4+284672*F1^5*F3^5-188416*F1^3*F2*F3^5+24576*F1^2*F3^6-720*F1^16*F4+11200*F1^14*F2*F4-66560*F1^12*F2^2*F4+184320*F1^10*F2^3*F4-225280*F1^8*F2^4*F4+81920*F1^6*F2^5*F4-19200*F1^13*F3*F4+222720*F1^11*F2*F3*F4-890880*F1^9*F2^2*F3*F4+1351680*F1^7*F2^3*F3*F4-491520*F1^5*F2^4*F3*F4-188160*F1^10*F3^2*F4+1449984*F1^8*F2*F3^2*F4-3059712*F1^6*F2^2*F3^2*F4+1081344*F1^4*F2^3*F3^2*F4-798720*F1^7*F3^3*F4+3129344*F1^5*F2*F3^3*F4-1048576*F1^3*F2^2*F3^3*F4-1234944*F1^4*F3^4*F4+417792*F1^2*F2*F3^4*F4-49152*F1*F3^5*F4+8064*F1^12*F4^2-89088*F1^10*F2*F4^2+331776*F1^8*F2^2*F4^2-442368*F1^6*F2^3*F4^2+98304*F1^4*F2^4*F4^2+153600*F1^9*F3*F4^2-1105920*F1^7*F2*F3*F4^2+2064384*F1^5*F2^2*F3*F4^2-393216*F1^3*F2^3*F3*F4^2+933888*F1^6*F3^2*F4^2-3268608*F1^4*F2*F3^2*F4^2+491520*F1^2*F2^2*F3^2*F4^2+1769472*F1^3*F3^3*F4^2-196608*F1*F2*F3^3*F4^2+16384*F3^4*F4^2-30720*F1^8*F4^3+196608*F1^6*F2*F4^3-294912*F1^4*F2^2*F4^3-344064*F1^5*F3*F4^3+983040*F1^3*F2*F3*F4^3-819200*F1^2*F3^2*F4^3+16384*F1^4*F4^4+320*F1^15*F5-4480*F1^13*F2*F5+23040*F1^11*F2^2*F5-51200*F1^9*F2^3*F5+40960*F1^7*F2^4*F5+10752*F1^12*F3*F5-113664*F1^10*F2*F3*F5+399360*F1^8*F2^2*F3*F5-491520*F1^6*F2^3*F3*F5+98304*F1^4*F2^4*F3*F5+124416*F1^9*F3^2*F5-872448*F1^7*F2*F3^2*F5+1597440*F1^5*F2^2*F3^2*F5-393216*F1^3*F2^3*F3^2*F5+593920*F1^6*F3^3*F5-2072576*F1^4*F2*F3^3*F5+491520*F1^2*F2^2*F3^3*F5+983040*F1^3*F3^4*F5-196608*F1*F2*F3^4*F5+16384*F3^5*F5-7680*F1^11*F4*F5+73728*F1^9*F2*F4*F5-221184*F1^7*F2^2*F4*F5+196608*F1^5*F2^3*F4*F5-184320*F1^8*F3*F4*F5+1155072*F1^6*F2*F3*F4*F5-1720320*F1^4*F2^2*F3*F4*F5+196608*F1^2*F2^3*F3*F4*F5-1302528*F1^5*F3^2*F4*F5+3833856*F1^3*F2*F3^2*F4*F5-393216*F1*F2^2*F3^2*F4*F5-2654208*F1^2*F3^3*F4*F5+131072*F2*F3^3*F4*F5+49152*F1^7*F4^2*F5-245760*F1^5*F2*F4^2*F5+196608*F1^3*F2^2*F4^2*F5+671744*F1^4*F3*F4^2*F5-1310720*F1^2*F2*F3*F4^2*F5+1638400*F1*F3^2*F4^2*F5-65536*F1^3*F4^3*F5-11520*F1^10*F5^2+118272*F1^8*F2*F5^2-417792*F1^6*F2^2*F5^2+565248*F1^4*F2^3*F5^2-196608*F1^2*F2^4*F5^2-110592*F1^7*F3*F5^2+712704*F1^5*F2*F3*F5^2-1277952*F1^3*F2^2*F3*F5^2+393216*F1*F2^3*F3*F5^2-118784*F1^4*F3^2*F5^2+376832*F1^2*F2*F3^2*F5^2-131072*F2^2*F3^2*F5^2+557056*F1*F3^3*F5^2+135168*F1^6*F4*F5^2-688128*F1^4*F2*F4*F5^2+786432*F1^2*F2^2*F4*F5^2+491520*F1^3*F3*F4*F5^2-917504*F1*F2*F3*F4*F5^2-491520*F3^2*F4*F5^2-229376*F1^2*F4^2*F5^2-172032*F1^5*F5^3+688128*F1^3*F2*F5^3-393216*F1*F2^2*F5^3-819200*F1^2*F3*F5^3+589824*F2*F3*F5^3+589824*F1*F4*F5^3-442368*F5^4)+16*S^2*(7*F1^24-168*F1^22*F2+1704*F1^20*F2^2-9440*F1^18*F2^3+30720*F1^16*F2^4-58368*F1^14*F2^5+59392*F1^12*F2^6-24576*F1^10*F2^7+288*F1^21*F3-5760*F1^19*F2*F3+47040*F1^17*F2^2*F3-199680*F1^15*F2^3*F3+460800*F1^13*F2^4*F3-540672*F1^11*F2^5*F3+245760*F1^9*F2^6*F3+4880*F1^18*F3^2-78240*F1^16*F2*F3^2+486400*F1^14*F2^2*F3^2-1448960*F1^12*F2^3*F3^2+2027520*F1^10*F2^4*F3^2-1024000*F1^8*F2^5*F3^2+43520*F1^15*F3^3-527360*F1^13*F2*F3^3+2273280*F1^11*F2^2*F3^3-4014080*F1^9*F2^3*F3^3+2293760*F1^7*F2^4*F3^3+215296*F1^12*F3^4-1784832*F1^10*F2*F3^4+4435968*F1^8*F2^2*F3^4-2965504*F1^6*F2^3*F3^4+563200*F1^9*F3^5-2605056*F1^7*F2*F3^5+2195456*F1^5*F2^2*F3^5+638976*F1^6*F3^6-851968*F1^4*F2*F3^6+131072*F1^3*F3^7-288*F1^20*F4+5600*F1^18*F2*F4-44160*F1^16*F2^2*F4+179200*F1^14*F2^3*F4-389120*F1^12*F2^4*F4+417792*F1^10*F2^5*F4-163840*F1^8*F2^6*F4-9600*F1^17*F3*F4+148480*F1^15*F2*F3*F4-880640*F1^13*F2^2*F3*F4+2457600*F1^11*F2^3*F3*F4-3112960*F1^9*F2^4*F3*F4+1310720*F1^7*F2^5*F3*F4-125440*F1^14*F3^2*F4+1447936*F1^12*F2*F3^2*F4-5824512*F1^10*F2^2*F3^2*F4+9207808*F1^8*F2^3*F3^2*F4-4194304*F1^6*F2^4*F3^2*F4-798720*F1^11*F3^3*F4+6160384*F1^9*F2*F3^3*F4-13565952*F1^7*F2^2*F3^3*F4+6815744*F1^5*F2^3*F3^3*F4-2463744*F1^8*F3^4*F4+10010624*F1^6*F2*F3^4*F4-5865472*F1^4*F2^2*F3^4*F4-2981888*F1^5*F3^5*F4+2490368*F1^3*F2*F3^5*F4-393216*F1^2*F3^6*F4+4096*F1^16*F4^2-60416*F1^14*F2*F4^2+335872*F1^12*F2^2*F4^2-851968*F1^10*F2^3*F4^2+917504*F1^8*F2^4*F4^2-262144*F1^6*F2^5*F4^2+104448*F1^13*F3*F4^2-1130496*F1^11*F2*F3*F4^2+4128768*F1^9*F2^2*F3*F4^2-5505024*F1^7*F2^3*F3*F4^2+1572864*F1^5*F2^4*F3*F4^2+958464*F1^10*F3^2*F4^2-6709248*F1^8*F2*F3^2*F4^2+12386304*F1^6*F2^2*F3^2*F4^2-3538944*F1^4*F2^3*F3^2*F4^2+3670016*F1^7*F3^3*F4^2-12451840*F1^5*F2*F3^3*F4^2+3670016*F1^3*F2^2*F3^3*F4^2+4751360*F1^4*F3^4*F4^2-1703936*F1^2*F2*F3^4*F4^2+262144*F1*F3^5*F4^2-22528*F1^12*F4^3+221184*F1^10*F2*F4^3-688128*F1^8*F2^2*F4^3+655360*F1^6*F2^3*F4^3-393216*F1^9*F3*F4^3+2359296*F1^7*F2*F3*F4^3-3145728*F1^5*F2^2*F3*F4^3-2031616*F1^6*F3^2*F4^3+4980736*F1^4*F2*F3^2*F4^3-2621440*F1^3*F3^3*F4^3+32768*F1^8*F4^4-131072*F1^6*F2*F4^4+262144*F1^5*F3*F4^4+160*F1^19*F5-2880*F1^17*F2*F5+20480*F1^15*F2^2*F5-71680*F1^13*F2^3*F5+122880*F1^11*F2^4*F5-81920*F1^9*F2^5*F5+6528*F1^16*F3*F5-94208*F1^14*F2*F3*F5+507904*F1^12*F2^2*F3*F5-1228800*F1^10*F2^3*F3*F5+1212416*F1^8*F2^4*F3*F5-262144*F1^6*F2^5*F3*F5+99328*F1^13*F3^2*F5-1069056*F1^11*F2*F3^2*F5+3883008*F1^9*F2^2*F3^2*F5-5177344*F1^7*F2^3*F3^2*F5+1572864*F1^5*F2^4*F3^2*F5+708608*F1^10*F3^3*F5-5038080*F1^8*F2*F3^3*F5+9699328*F1^6*F2^2*F3^3*F5-3538944*F1^4*F2^3*F3^3*F5+2359296*F1^7*F3^4*F5-8519680*F1^5*F2*F3^4*F5+3670016*F1^3*F2^2*F3^4*F5+2916352*F1^4*F3^5*F5-1703936*F1^2*F2*F3^5*F5+262144*F1*F3^6*F5-5120*F1^15*F4*F5+69632*F1^13*F2*F4*F5-344064*F1^11*F2^2*F4*F5+720896*F1^9*F2^3*F4*F5-524288*F1^7*F2^4*F4*F5-159744*F1^12*F3*F4*F5+1597440*F1^10*F2*F3*F4*F5-5160960*F1^8*F2^2*F3*F4*F5+5505024*F1^6*F2^3*F3*F4*F5-786432*F1^4*F2^4*F3*F4*F5-1695744*F1^9*F3^2*F4*F5+10813440*F1^7*F2*F3^2*F4*F5-16908288*F1^5*F2^2*F3^2*F4*F5+3145728*F1^3*F2^3*F3^2*F4*F5-7143424*F1^6*F3^3*F4*F5+20971520*F1^4*F2*F3^3*F4*F5-4194304*F1^2*F2^2*F3^3*F4*F5-9437184*F1^3*F3^4*F4*F5+2097152*F1*F2*F3^4*F4*F5-262144*F3^5*F4*F5+49152*F1^11*F4^2*F5-442368*F1^9*F2*F4^2*F5+1179648*F1^7*F2^2*F4^2*F5-786432*F1^5*F2^3*F4^2*F5+1048576*F1^8*F3*F4^2*F5-5636096*F1^6*F2*F3*F4^2*F5+5767168*F1^4*F2^2*F3*F4^2*F5+6160384*F1^5*F3^2*F4^2*F5-12058624*F1^3*F2*F3^2*F4^2*F5+7864320*F1^2*F3^3*F4^2*F5-131072*F1^7*F4^3*F5+524288*F1^5*F2*F4^3*F5-1310720*F1^4*F3*F4^3*F5-7424*F1^14*F5^2+105984*F1^12*F2*F5^2-583680*F1^10*F2^2*F5^2+1531904*F1^8*F2^3*F5^2-1867776*F1^6*F2^4*F5^2+786432*F1^4*F2^5*F5^2-104448*F1^11*F3*F5^2+1093632*F1^9*F2*F3*F5^2-4128768*F1^7*F2^2*F3*F5^2+6488064*F1^5*F2^3*F3*F5^2-3145728*F1^3*F2^4*F3*F5^2-280576*F1^8*F3^2*F5^2+2146304*F1^6*F2*F3^2*F5^2-5931008*F1^4*F2^2*F3^2*F5^2+4194304*F1^2*F2^3*F3^2*F5^2+1015808*F1^5*F3^3*F5^2-1179648*F1^3*F2*F3^3*F5^2-2097152*F1*F2^2*F3^3*F5^2+2883584*F1^2*F3^4*F5^2+262144*F2*F3^4*F5^2+129024*F1^10*F4*F5^2-1105920*F1^8*F2*F4*F5^2+3047424*F1^6*F2^2*F4*F5^2-2752512*F1^4*F2^3*F4*F5^2+884736*F1^7*F3*F4*F5^2-4194304*F1^5*F2*F3*F4*F5^2+5767168*F1^3*F2^2*F3*F4*F5^2-1146880*F1^4*F3^2*F4*F5^2+393216*F1^2*F2*F3^2*F4*F5^2-4980736*F1*F3^3*F4*F5^2-557056*F1^6*F4^2*F5^2+1441792*F1^4*F2*F4^2*F5^2-262144*F1^3*F3*F4^2*F5^2-233472*F1^9*F5^3+1867776*F1^7*F2*F5^3-4521984*F1^5*F2^2*F5^3+3145728*F1^3*F2^3*F5^3-2326528*F1^6*F3*F5^3+9895936*F1^4*F2*F3*F5^3-8650752*F1^2*F2^2*F3*F5^3-4718592*F1^3*F3^2*F5^3+6815744*F1*F2*F3^2*F5^3-262144*F3^3*F5^3+2162688*F1^5*F4*F5^3-5505024*F1^3*F2*F4*F5^3+6029312*F1^2*F3*F4*F5^3-1572864*F1^4*F5^4+3145728*F1^2*F2*F5^4-4718592*F1*F3*F5^4)+F1*(F1^5-4*F1^3*F2+8*F1^2*F3-16*F1*F4+32*F5)*(F1^6-6*F1^4*F2+8*F1^2*F2^2+8*F1^3*F3-16*F1*F2*F3+8*F3^2-8*F1^2*F4+16*F1*F5)^2*(F1^10-12*F1^8*F2+48*F1^6*F2^2-64*F1^4*F2^3+24*F1^7*F3-192*F1^5*F2*F3+384*F1^3*F2^2*F3+192*F1^4*F3^2-768*F1^2*F2*F3^2+512*F1*F3^3-16*F1^6*F4+128*F1^4*F2*F4-256*F1^2*F2^2*F4-256*F1^3*F3*F4+1024*F1*F2*F3*F4-1024*F3^2*F4-32*F1^5*F5+256*F1^3*F2*F5-512*F1*F2^2*F5-256*F1^2*F3*F5+1024*F2*F3*F5-1024*F5^2)


F1=a+b+c+d+e
F2=a⋅(b+c+d+e)+b⋅(c+d+e)+c⋅(d+e)+d⋅e
F3=a⋅b⋅(c+d+e)+a⋅c⋅(d+e)+a⋅d⋅e+b⋅c⋅(d+e)+b⋅d⋅e+c⋅d⋅e
F4=a⋅b⋅c⋅(d+e)+a⋅b⋅d⋅e+a⋅c⋅d⋅e+b⋅c⋅d⋅e
F5=a⋅b⋅c⋅d⋅e

用Fn替换，不用s避免呼啸

数学星空

经验算，楼上的结果是正确的：

例如取：\(a=1,b=1,c=1,d=1,e=1\),可以得到\(s=\frac{\sqrt{-10\sqrt{5}+25}}{4}\)

取\(a=1,b=2,c=3,d=4,e=5\),由51#得到

\(-28733079375R^{14}+663071062500R^{12}-5537171441250R^{10}+19500103012500R^8-23120125194375R^6-7138513800000R^4+25147584000000R^2+2985984000000, -28733079375R^{14}+663071062500R^{12}-5537171441250R^{10}+19500103012500R^8-23120125194375R^6-7138513800000R^4+25147584000000R^2+2985984000000=0\)

取实根\(R=2.71756722526196\)

再代入54#得到：

2.245432996*10^37-1.1264*10^38*s^2+3.0134272*10^38*s^4-3.068579021*10^38*s^6+1.651588989*10^38*s^8-5.050060631*10^37*s^10+9.158471895*10^36*s^12-1.038263487*10^36*s^14+7.677398631*10^34*s^16-3.812226838*10^33*s^18+1.292421408*10^32*s^20-3.004500060*10^30*s^22+4.745647333*10^28*s^24-4.958837152*10^26*s^26+3.246169511*10^24*s^28-1.191175915*10^22*s^30+18446744073709551616*s^32＝0
得到：\(s=13.6049923845358\)

＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝
由楼上的结论可得直接得到：

\(268435456s^{14}-56656658432s^{12}+411349024768s^{10}+163698584518656s^8-85159641600000s^6-19121610163680000s^4-68707423496250000s^2-176008069346484375＝0\)

也可以算得：\(s=13.6049923838726\)

＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝
当然最直接的可以由下列方程确定结果是否准确

\(\arcsin(\frac{a}{2R})+\arcsin(\frac{b}{2R})+\arcsin(\frac{c}{2R})+\arcsin(\frac{d}{2R})+\arcsin(\frac{e}{2R})-\pi=0\)

代入\(a=1,b=2,c=3,d=4,e=5\) 得到\(R=2.71756722502368\)

\(a\sqrt{4R^2-a^2}+b\sqrt{4R^2-b^2}+c\sqrt{4R^2-c^2}+d\sqrt{4R^2-d^2}+e\sqrt{4R^2-e^2}-4s=0\)

容易算得：\(s=13.6049923808433\)

数学星空

TO  zeroieme :

请问你使用什么软件和程序消元得到56#的结果？

若有兴趣，你试着给出下面两个问题的答案？

1. http://bbs.emath.ac.cn/forum.php ... 12&fromuid=1455


2. http://bbs.emath.ac.cn/forum.php ... 67&fromuid=1455  类似的结果。

对于双椭圆（内接和外切)的七边形\(ABCDEFG\)(七边依次为$a,b,c,d,e,f,g$),存在的条件？

内接椭圆为$x^2/m^2+y^2/n^2=1$

$(m\cos(t)-mcos(t+(2\pi)/7))^2+(n\sin(t)-n\sin(t+(2\pi)/7))^2=a^2$

$(m\cos(t+(2\pi)/7)-m\cos(t+(4\pi)/7))^2+(nsin(t+(2\pi)/7)-nsin(t+(4\pi)/7))^2=b^2 $

$(m\cos(t+(4\pi)/7)-m\cos(t+(6\pi)/7))^2+(n\sin(t+(4\pi)/7)-n\sin(t+(6\pi)/7))^2=c^2 $

$(m\cos(t+(6\pi)/7)-m\cos(t+(8\pi)/7))^2+(n\sin(t+(6\pi)/7)-n\sin(t+(8\pi)/7))^2=d^2$

$(m\cos(t+(8\pi)/7)-m\cos(t+(10\pi)/7))^2+(n\sin(t+(8\pi)/7)-n\sin(t+(10\pi)/7))^2=e^2$

$(m\cos(t+(10\pi)/7)-m\cos(t+(12\pi)/7))^2+(n\sin(t+(10\pi)/7)-n\sin(t+(12\pi)/7))^2=f^2$

$(m\cos(t+(12\pi)/7)-m\cos(t+2\pi))^2+(n\sin(t+(12\pi)/7)-n\sin(t+2\pi))^2=g^2$

给出\(a,b,c,d,e,f,g\)之间的关系式？

zeroieme

数学星空 发表于 2014-4-19 02:05
经验算，楼上的结果是正确的：

例如取：\(a=1,b=1,c=1,d=1,e=1\),可以得到\(s=\frac{\sqrt{-10\sqrt{5}+ ...

就是用mathematica 9的Resultant、PolynomialRemainder算的。
看26楼，本来只用Resultant，发现它会引入不符合原意的因式，所以加上PolynomialRemainder辗转求余。

zeroieme

假定方程是P=0形式，以下我只写多项式部分P
4个多项式
$a^4 r^2-2 a^2 b^2 r^2+b^4 r^2+a^2 b^2 x^2-2 a^2 r^2 x^2-2 b^2 r^2 x^2+r^2 x^4$
$c^4 r^2-2 c^2 r^2 x^2+r^2 x^4-2 c^2 r^2 y^2+c^2 x^2 y^2-2 r^2 x^2 y^2+r^2 y^4$
$d^4 r^2-2 d^2 e^2 r^2+e^4 r^2+d^2 e^2 y^2-2 d^2 r^2 y^2-2 e^2 r^2 y^2+r^2 y^4$
$4 r S-a b x-d e y-c x y$

首先是 2、3式Resultant消去y，以及4式解一次方程代入2、3式消去y。