三角形两内点间距

creasson

以前我写过一篇小短文，对于你所提的绝大多数几何问题，是可以容易解决的。后来知道国外也早有人用重心坐标解决几何问题的，并且给出了几千个点的重心坐标，可惜始终没有流行开来。无毒史那一套建立在复数+仿射方法上的系统，大约也是正确的，但是他的东西实在古怪，运算过程又隐于软件中，加上他梦呓般说话显得古老，倒是乏人问津。无论怎样，都可以看到一个很明确的趋势，几何是在与代数融合。解析几何，三角函数，复数，射影几何都是将几何对象转化成代数。代数的发展状况直接制约着几何的发展。当然，几何直观上的优势还是有的，能给人以灵感上的启迪创造，但是就初等几何而言，似乎已经很有限了。
面积坐标与体积坐标之应用.pdf (127.8 KB, 下载次数: 42)

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文中一些论述可能有失偏颇，大约是令人不快的，这仅是个人的观点，各位倒无须放在心上，两日后我会删除这段回复。

葡萄糖

数学星空 发表于 2014-2-17 20:17
我们可以利用‘万能心距公式’计算‘陈都提出的完全心距公式（他得到的公式都很美）’

1.  三个旁心\(O_1,O_2,O_3\)
2.  费马点\(M\)

关于费马点与重心的距离公式

http://wenku.baidu.com/link?url= ... SJ6icK5-_kFj_pF_mHu

葡萄糖

葡萄糖 发表于 2014-2-22 11:03
关于费马点与重心的距离公式

关于费马点与其他心的距离公式？

数学星空

关于三个旁心（\(O_1,O_2,O_3\)),我们得到如下结果：

1. \(a^6+b^6+c^6+2a^3b^2c-2a^2b^3c-2a^2bc^3+2ab^3c^2-3abc^4+6a^2b^2c^2+3a^4bc-3ab^4c+2ab^2c^3+2a^3bc^2+a^5b+a^5c-a^4b^2-a^4c^2-2a^3b^3-2a^3c^3-a^2b^4-a^2c^4+ab^5+ac^5-b^5c-b^4c^2+2b^3c^3-b^2c^4-bc^5+(c+a+b)(a+b-c)(a-b-c)(a-b+c)HO_1^2=0\)

2.\((9a-9b-9c)GO_1^2+a^3+2a^2b+2a^2c-2ab^2+9abc-2ac^2-b^3+2b^2c+2bc^2-c^3=0\)

3.\((a+b-c)(c+a+b)(a-b+c)(a-b-c)OO_1^2+abc(a^3+a^2b+a^2c-ab^2+3abc-ac^2-b^3+b^2c+bc^2-c^3)=0\)

4.\((c+a+b)(a-b-c)IO_1^2+4a^2bc=0\)

5.\((a-b-c)(a-b+c)O_1O_2^2+4abc^2=0\)

6.\((a-b-c)(a+b-c)O_1O_3^2+4ab^2c=0\)

7.\((a+b-c)(c+a+b)(a-b+c)(a-b-c)HO_2^2+a^6+b^6+c^6-2a^3b^2c+2a^2b^3c+2a^2bc^3+2ab^3c^2-3abc^4+6a^2b^2c^2-3a^4bc+3ab^4c-2ab^2c^3+2a^3bc^2+a^5b-a^5c-a^4b^2-a^4c^2-2a^3b^3+2a^3c^3-a^2b^4-a^2c^4+ab^5-ac^5+b^5c-b^4c^2-2b^3c^3-b^2c^4+bc^5=0\)

8.\((9a-9b+9c)GO_2^2+a^3+2a^2b-2a^2c-2ab^2-9abc-2ac^2-b^3-2b^2c+2bc^2+c^3=0\)

9.\((a+b-c)(c+a+b)(a-b+c)(a-b-c)OO_2^2-abc(a^3+a^2b-a^2c-ab^2-3abc-ac^2-b^3-b^2c+bc^2+c^3)=0\)

10.\((c+a+b)(a-b+c)IO_2^2-4ab^2c=0\)

11.\((a+b-c)(c+a+b)(a-b+c)(a-b-c)HO_3^2+a^6+b^6+c^6+2a^3b^2c+2a^2b^3c+2a^2bc^3-2ab^3c^2+3abc^4+6a^2b^2c^2-3a^4bc-3ab^4c+2ab^2c^3-2a^3bc^2-a^5b+a^5c-a^4b^2-a^4c^2+2a^3b^3-2a^3c^3-a^2b^4-a^2c^4-ab^5+ac^5+b^5c-b^4c^2-2b^3c^3-b^2c^4+bc^5=0\)

12.\(9a+9b-9c)GO_3^2+a^3-2a^2b+2a^2c-2ab^2-9abc-2ac^2+b^3+2b^2c-2bc^2-c^3=0\)

13.\((a+b-c)(c+a+b)(a-b+c)(a-b-c)OO_3^2-abc(a^3-a^2b+a^2c-ab^2-3abc-ac^2+b^3+b^2c-bc^2-c^3)=0\)

14.\((a+b-c)(c+a+b)IO_3^2-4abc^2=0\)

15.\((a+b-c)(a-b+c)O_2O_3^2-4a^2bc=0\)

数学星空

若令\(a=y+z,b=x+z,c=x+y\),上面的结果可以化为：

1.\(x^4y^2-2x^4yz+x^4z^2+2x^3y^3-2x^3y^2z-2x^3yz^2+2x^3z^3+x^2y^4-2x^2y^3z+6x^2y^2z^2-2x^2yz^3+x^2z^4-2xy^4z+14xy^3z^2+14xy^2z^3-2xyz^4+9y^4z^2+18y^3z^3+9y^2z^4-4(x+y+z)HO_1^2xyz=0\)

2. \(-9GO_1^2x+x^3+4x^2y+4x^2z+4xy^2+13xyz+4xz^2+9y^2z+9yz^2=0\)

3.\(-16(x+y+z)xyz OO_1^2+(y+z)(x+z)(x+y)(x^2y+x^2z+xy^2+10xyz+xz^2+9y^2z+9yz^2)=0\)

4.\(-(x+y+z)x IO_1^2+(y+z)^2(x+z)(x+y)=0\)

5.\(-O_1O_2^2 xy+(y+z)(x+z)(x+y)^2=0\)

6.\(-O_1O_3^2 xz+(y+z)(x+z)^2(x+y)=0\)

7.\(x^4y^2-2x^4yz+9x^4z^2+2x^3y^3-2x^3y^2z+14x^3yz^2+18x^3z^3+x^2y^4-2x^2y^3z+6x^2y^2z^2+14x^2yz^3+9x^2z^4-2xy^4z-2xy^3z^2-2xy^2z^3-2xyz^4+y^4z^2+2y^3z^3+y^2z^4-4(x+y+z)HO_2^2xyz=0\)

8.\(9GO_2^2y-4x^2y-9x^2z-4xy^2-13xyz-9xz^2-y^3-4y^2z-4yz^2=0\)

9.\(-16(x+y+z)OO_2^2 xyz+(y+z)(x+z)(x+y)(x^2y+9x^2z+xy^2+10xyz+9xz^2+y^2z+yz^2)=0\)

10.\( (x+y+z)IO_2^2y-(y+z)(x+z)^2(x+y)=0\)

11.\(9x^4y^2-2x^4yz+x^4z^2+18x^3y^3+14x^3y^2z-2x^3yz^2+2x^3z^3+9x^2y^4+14x^2y^3z+6x^2y^2z^2-2x^2yz^3+x^2z^4-2xy^4z-2xy^3z^2-2xy^2z^3-2xyz^4+y^4z^2+2y^3z^3+y^2z^4-4(x+y+z)HO_3^2xyz=0\)

12.\(9GO_3^2z-9x^2y-4x^2z-9xy^2-13xyz-4xz^2-4y^2z-4yz^2-z^3=0\)

13.\(-16(x+y+z)OO_3^2 xyz+(y+z)(x+z)(x+y)(9x^2y+x^2z+9xy^2+10xyz+xz^2+y^2z+yz^2)=0\)

14.\((x+y+z)zIO_3^2-(y+z)(x+z)(x+y)^2=0\)

15.\(O_2O_3^2 yz-(y+z)^2(x+z)(x+y)=0\)

数学星空

关于费马点\(F\)的结果：（仅讨论\( \max{A,B,C}<120^\circ \)情形）
1.   81*(a^2+b^2+c^2)^2*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)^2*(a+b-c)^4*(a+b+c)^4*(a-b+c)^4*(a-b-c)^4*OF^8+(27*(a^2+b^2+c^2))*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(2*a^12-7*a^10*b^2-7*a^10*c^2+10*a^8*b^4+22*a^8*b^2*c^2+10*a^8*c^4-10*a^6*b^6-3*a^6*b^4*c^2-3*a^6*b^2*c^4-10*a^6*c^6+10*a^4*b^8-3*a^4*b^6*c^2-42*a^4*b^4*c^4-3*a^4*b^2*c^6+10*a^4*c^8-7*a^2*b^10+22*a^2*b^8*c^2-3*a^2*b^6*c^4-3*a^2*b^4*c^6+22*a^2*b^2*c^8-7*a^2*c^10+2*b^12-7*b^10*c^2+10*b^8*c^4-10*b^6*c^6+10*b^4*c^8-7*b^2*c^10+2*c^12)*(a+b-c)^3*(a+b+c)^3*(a-b+c)^3*(a-b-c)^3*OF^6+(27*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a^20-5*a^18*b^2-5*a^18*c^2+10*a^16*b^4+23*a^16*b^2*c^2+10*a^16*c^4-10*a^14*b^6-37*a^14*b^4*c^2-37*a^14*b^2*c^4-10*a^14*c^6+5*a^12*b^8+23*a^12*b^6*c^2+28*a^12*b^4*c^4+23*a^12*b^2*c^6+5*a^12*c^8-2*a^10*b^10-4*a^10*b^8*c^2+43*a^10*b^6*c^4+43*a^10*b^4*c^6-4*a^10*b^2*c^8-2*a^10*c^10+5*a^8*b^12-4*a^8*b^10*c^2-16*a^8*b^8*c^4-84*a^8*b^6*c^6-16*a^8*b^4*c^8-4*a^8*b^2*c^10+5*a^8*c^12-10*a^6*b^14+23*a^6*b^12*c^2+43*a^6*b^10*c^4-84*a^6*b^8*c^6-84*a^6*b^6*c^8+43*a^6*b^4*c^10+23*a^6*b^2*c^12-10*a^6*c^14+10*a^4*b^16-37*a^4*b^14*c^2+28*a^4*b^12*c^4+43*a^4*b^10*c^6-16*a^4*b^8*c^8+43*a^4*b^6*c^10+28*a^4*b^4*c^12-37*a^4*b^2*c^14+10*a^4*c^16-5*a^2*b^18+23*a^2*b^16*c^2-37*a^2*b^14*c^4+23*a^2*b^12*c^6-4*a^2*b^10*c^8-4*a^2*b^8*c^10+23*a^2*b^6*c^12-37*a^2*b^4*c^14+23*a^2*b^2*c^16-5*a^2*c^18+b^20-5*b^18*c^2+10*b^16*c^4-10*b^14*c^6+5*b^12*c^8-2*b^10*c^10+5*b^8*c^12-10*b^6*c^14+10*b^4*c^16-5*b^2*c^18+c^20)*(a+b-c)^2*(a+b+c)^2*(a-b+c)^2*(a-b-c)^2*OF^4+(3*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(2*a^26-16*a^24*b^2-16*a^24*c^2+54*a^22*b^4+120*a^22*b^2*c^2+54*a^22*c^4-98*a^20*b^6-351*a^20*b^4*c^2-351*a^20*b^2*c^4-98*a^20*c^6+100*a^18*b^8+490*a^18*b^6*c^2+852*a^18*b^4*c^4+490*a^18*b^2*c^6+100*a^18*c^8-54*a^16*b^10-240*a^16*b^8*c^2-966*a^16*b^6*c^4-966*a^16*b^4*c^6-240*a^16*b^2*c^8-54*a^16*c^10+12*a^14*b^12-258*a^14*b^10*c^2+486*a^14*b^8*c^4+830*a^14*b^6*c^6+486*a^14*b^4*c^8-258*a^14*b^2*c^10+12*a^14*c^12+12*a^12*b^14+510*a^12*b^12*c^2-75*a^12*b^10*c^4-232*a^12*b^8*c^6-232*a^12*b^6*c^8-75*a^12*b^4*c^10+510*a^12*b^2*c^12+12*a^12*c^14-54*a^10*b^16-258*a^10*b^14*c^2-75*a^10*b^12*c^4+384*a^10*b^10*c^6-402*a^10*b^8*c^8+384*a^10*b^6*c^10-75*a^10*b^4*c^12-258*a^10*b^2*c^14-54*a^10*c^16+100*a^8*b^18-240*a^8*b^16*c^2+486*a^8*b^14*c^4-232*a^8*b^12*c^6-402*a^8*b^10*c^8-402*a^8*b^8*c^10-232*a^8*b^6*c^12+486*a^8*b^4*c^14-240*a^8*b^2*c^16+100*a^8*c^18-98*a^6*b^20+490*a^6*b^18*c^2-966*a^6*b^16*c^4+830*a^6*b^14*c^6-232*a^6*b^12*c^8+384*a^6*b^10*c^10-232*a^6*b^8*c^12+830*a^6*b^6*c^14-966*a^6*b^4*c^16+490*a^6*b^2*c^18-98*a^6*c^20+54*a^4*b^22-351*a^4*b^20*c^2+852*a^4*b^18*c^4-966*a^4*b^16*c^6+486*a^4*b^14*c^8-75*a^4*b^12*c^10-75*a^4*b^10*c^12+486*a^4*b^8*c^14-966*a^4*b^6*c^16+852*a^4*b^4*c^18-351*a^4*b^2*c^20+54*a^4*c^22-16*a^2*b^24+120*a^2*b^22*c^2-351*a^2*b^20*c^4+490*a^2*b^18*c^6-240*a^2*b^16*c^8-258*a^2*b^14*c^10+510*a^2*b^12*c^12-258*a^2*b^10*c^14-240*a^2*b^8*c^16+490*a^2*b^6*c^18-351*a^2*b^4*c^20+120*a^2*b^2*c^22-16*a^2*c^24+2*b^26-16*b^24*c^2+54*b^22*c^4-98*b^20*c^6+100*b^18*c^8-54*b^16*c^10+12*b^14*c^12+12*b^12*c^14-54*b^10*c^16+100*b^8*c^18-98*b^6*c^20+54*b^4*c^22-16*b^2*c^24+2*c^26)*(a+b-c)*(a+b+c)*(a-b+c)*(a-b-c)*OF^2+(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(a^16-5*a^14*b^2-5*a^14*c^2+10*a^12*b^4+19*a^12*b^2*c^2+10*a^12*c^4-11*a^10*b^6-27*a^10*b^4*c^2-27*a^10*b^2*c^4-11*a^10*c^6+10*a^8*b^8+13*a^8*b^6*c^2+33*a^8*b^4*c^4+13*a^8*b^2*c^6+10*a^8*c^8-11*a^6*b^10+13*a^6*b^8*c^2-23*a^6*b^6*c^4-23*a^6*b^4*c^6+13*a^6*b^2*c^8-11*a^6*c^10+10*a^4*b^12-27*a^4*b^10*c^2+33*a^4*b^8*c^4-23*a^4*b^6*c^6+33*a^4*b^4*c^8-27*a^4*b^2*c^10+10*a^4*c^12-5*a^2*b^14+19*a^2*b^12*c^2-27*a^2*b^10*c^4+13*a^2*b^8*c^6+13*a^2*b^6*c^8-27*a^2*b^4*c^10+19*a^2*b^2*c^12-5*a^2*c^14+b^16-5*b^14*c^2+10*b^12*c^4-11*b^10*c^6+10*b^8*c^8-11*b^6*c^10+10*b^4*c^12-5*b^2*c^14+c^16)*(a^16-5*a^14*b^2-5*a^14*c^2+10*a^12*b^4+28*a^12*b^2*c^2+10*a^12*c^4-11*a^10*b^6-54*a^10*b^4*c^2-54*a^10*b^2*c^4-11*a^10*c^6+10*a^8*b^8+31*a^8*b^6*c^2+51*a^8*b^4*c^4+31*a^8*b^2*c^6+10*a^8*c^8-11*a^6*b^10+31*a^6*b^8*c^2+22*a^6*b^6*c^4+22*a^6*b^4*c^6+31*a^6*b^2*c^8-11*a^6*c^10+10*a^4*b^12-54*a^4*b^10*c^2+51*a^4*b^8*c^4+22*a^4*b^6*c^6+51*a^4*b^4*c^8-54*a^4*b^2*c^10+10*a^4*c^12-5*a^2*b^14+28*a^2*b^12*c^2-54*a^2*b^10*c^4+31*a^2*b^8*c^6+31*a^2*b^6*c^8-54*a^2*b^4*c^10+28*a^2*b^2*c^12-5*a^2*c^14+b^16-5*b^14*c^2+10*b^12*c^4-11*b^10*c^6+10*b^8*c^8-11*b^6*c^10+10*b^4*c^12-5*b^2*c^14+c^16)=0

2.   (6561*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a^2+b^2+c^2)^2*GF^8+(2916*(a^2+b^2+c^2))*(a^4-4*a^2*b^2-4*a^2*c^2+b^4-4*b^2*c^2+c^4)*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*GF^6+(243*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a^4-6*a^2*b^2-6*a^2*c^2+b^4-6*b^2*c^2+c^4)*(a^4-4*a^2*b^2-4*a^2*c^2+b^4-4*b^2*c^2+c^4)*GF^4-(18*(a^2+b^2+c^2))*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(a^4-4*a^2*b^2-4*a^2*c^2+b^4-4*b^2*c^2+c^4)^2*GF^2+(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)^2*(7*a^8-35*a^6*b^2-35*a^6*c^2+60*a^4*b^4+57*a^4*b^2*c^2+60*a^4*c^4-35*a^2*b^6+57*a^2*b^4*c^2+57*a^2*b^2*c^4-35*a^2*c^6+7*b^8-35*b^6*c^2+60*b^4*c^4-35*b^2*c^6+7*c^8)=0


3.   81*(a^2+b^2+c^2)^2*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)^2*(a+b-c)^4*(a+b+c)^4*(a-b+c)^4*(a-b-c)^4*HF^8+(54*(a^2+b^2+c^2))*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(4*a^12-11*a^10*b^2-11*a^10*c^2+20*a^8*b^4+20*a^8*b^2*c^2+20*a^8*c^4-26*a^6*b^6-9*a^6*b^4*c^2-9*a^6*b^2*c^4-26*a^6*c^6+20*a^4*b^8-9*a^4*b^6*c^2+6*a^4*b^4*c^4-9*a^4*b^2*c^6+20*a^4*c^8-11*a^2*b^10+20*a^2*b^8*c^2-9*a^2*b^6*c^4-9*a^2*b^4*c^6+20*a^2*b^2*c^8-11*a^2*c^10+4*b^12-11*b^10*c^2+20*b^8*c^4-26*b^6*c^6+20*b^4*c^8-11*b^2*c^10+4*c^12)*(a+b+c)^3*(a+b-c)^3*(a-b-c)^3*(a-b+c)^3*HF^6+(27*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(9*a^20-37*a^18*b^2-37*a^18*c^2+83*a^16*b^4+129*a^16*b^2*c^2+83*a^16*c^4-136*a^14*b^6-210*a^14*b^4*c^2-210*a^14*b^2*c^4-136*a^14*c^6+164*a^12*b^8+226*a^12*b^6*c^2+266*a^12*b^4*c^4+226*a^12*b^2*c^6+164*a^12*c^8-166*a^10*b^10-108*a^10*b^8*c^2-166*a^10*b^6*c^4-166*a^10*b^4*c^6-108*a^10*b^2*c^8-166*a^10*c^10+164*a^8*b^12-108*a^8*b^10*c^2+54*a^8*b^8*c^4+76*a^8*b^6*c^6+54*a^8*b^4*c^8-108*a^8*b^2*c^10+164*a^8*c^12-136*a^6*b^14+226*a^6*b^12*c^2-166*a^6*b^10*c^4+76*a^6*b^8*c^6+76*a^6*b^6*c^8-166*a^6*b^4*c^10+226*a^6*b^2*c^12-136*a^6*c^14+83*a^4*b^16-210*a^4*b^14*c^2+266*a^4*b^12*c^4-166*a^4*b^10*c^6+54*a^4*b^8*c^8-166*a^4*b^6*c^10+266*a^4*b^4*c^12-210*a^4*b^2*c^14+83*a^4*c^16-37*a^2*b^18+129*a^2*b^16*c^2-210*a^2*b^14*c^4+226*a^2*b^12*c^6-108*a^2*b^10*c^8-108*a^2*b^8*c^10+226*a^2*b^6*c^12-210*a^2*b^4*c^14+129*a^2*b^2*c^16-37*a^2*c^18+9*b^20-37*b^18*c^2+83*b^16*c^4-136*b^14*c^6+164*b^12*c^8-166*b^10*c^10+164*b^8*c^12-136*b^6*c^14+83*b^4*c^16-37*b^2*c^18+9*c^20)*(a+b+c)^2*(a+b-c)^2*(a-b-c)^2*(a-b+c)^2*HF^4+(6*(a^2+b^2+c^2))*(a+b+c)*(a+b-c)*(a-b-c)*(a-b+c)*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(a^8-a^6*b^2-a^6*c^2+a^4*b^2*c^2-a^2*b^6+a^2*b^4*c^2+a^2*b^2*c^4-a^2*c^6+b^8-b^6*c^2-b^2*c^6+c^8)*(13*a^16-53*a^14*b^2-53*a^14*c^2+94*a^12*b^4+160*a^12*b^2*c^2+94*a^12*c^4-107*a^10*b^6-162*a^10*b^4*c^2-162*a^10*b^2*c^4-107*a^10*c^6+106*a^8*b^8+55*a^8*b^6*c^2+42*a^8*b^4*c^4+55*a^8*b^2*c^6+106*a^8*c^8-107*a^6*b^10+55*a^6*b^8*c^2+52*a^6*b^6*c^4+52*a^6*b^4*c^6+55*a^6*b^2*c^8-107*a^6*c^10+94*a^4*b^12-162*a^4*b^10*c^2+42*a^4*b^8*c^4+52*a^4*b^6*c^6+42*a^4*b^4*c^8-162*a^4*b^2*c^10+94*a^4*c^12-53*a^2*b^14+160*a^2*b^12*c^2-162*a^2*b^10*c^4+55*a^2*b^8*c^6+55*a^2*b^6*c^8-162*a^2*b^4*c^10+160*a^2*b^2*c^12-53*a^2*c^14+13*b^16-53*b^14*c^2+94*b^12*c^4-107*b^10*c^6+106*b^8*c^8-107*b^6*c^10+94*b^4*c^12-53*b^2*c^14+13*c^16)*HF^2+(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(a^8-a^6*b^2-a^6*c^2+a^4*b^2*c^2-a^2*b^6+a^2*b^4*c^2+a^2*b^2*c^4-a^2*c^6+b^8-b^6*c^2-b^2*c^6+c^8)^2*(19*a^16-83*a^14*b^2-83*a^14*c^2+154*a^12*b^4+265*a^12*b^2*c^2+154*a^12*c^4-173*a^10*b^6-297*a^10*b^4*c^2-297*a^10*b^2*c^4-173*a^10*c^6+166*a^8*b^8+115*a^8*b^6*c^2+114*a^8*b^4*c^4+115*a^8*b^2*c^6+166*a^8*c^8-173*a^6*b^10+115*a^6*b^8*c^2+58*a^6*b^6*c^4+58*a^6*b^4*c^6+115*a^6*b^2*c^8-173*a^6*c^10+154*a^4*b^12-297*a^4*b^10*c^2+114*a^4*b^8*c^4+58*a^4*b^6*c^6+114*a^4*b^4*c^8-297*a^4*b^2*c^10+154*a^4*c^12-83*a^2*b^14+265*a^2*b^12*c^2-297*a^2*b^10*c^4+115*a^2*b^8*c^6+115*a^2*b^6*c^8-297*a^2*b^4*c^10+265*a^2*b^2*c^12-83*a^2*c^14+19*b^16-83*b^14*c^2+154*b^12*c^4-173*b^10*c^6+166*b^8*c^8-173*b^6*c^10+154*b^4*c^12-83*b^2*c^14+19*c^16)=0

4.   81*(a^2+b^2+c^2)^2*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)^2*(a+b+c)^4*IF^8+(27*(a^2+b^2+c^2))*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(2*a^9-a^8*b-a^8*c-6*a^7*b^2+12*a^7*b*c-6*a^7*c^2-18*a^6*b^2*c-18*a^6*b*c^2+5*a^5*b^4-16*a^5*b^2*c^2+5*a^5*c^4+5*a^4*b^5+14*a^4*b^4*c+20*a^4*b^3*c^2+20*a^4*b^2*c^3+14*a^4*b*c^4+5*a^4*c^5+20*a^3*b^4*c^2-36*a^3*b^3*c^3+20*a^3*b^2*c^4-6*a^2*b^7-18*a^2*b^6*c-16*a^2*b^5*c^2+20*a^2*b^4*c^3+20*a^2*b^3*c^4-16*a^2*b^2*c^5-18*a^2*b*c^6-6*a^2*c^7-a*b^8+12*a*b^7*c-18*a*b^6*c^2+14*a*b^4*c^4-18*a*b^2*c^6+12*a*b*c^7-a*c^8+2*b^9-b^8*c-6*b^7*c^2+5*b^5*c^4+5*b^4*c^5-6*b^2*c^7-b*c^8+2*c^9)*(a+b+c)^3*IF^6+(27*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a^14-a^13*b-a^13*c-4*a^12*b^2+6*a^12*b*c-4*a^12*c^2+3*a^11*b^3-9*a^11*b^2*c-9*a^11*b*c^2+3*a^11*c^3+6*a^10*b^4-5*a^10*b^3*c+31*a^10*b^2*c^2-5*a^10*b*c^3+6*a^10*c^4-3*a^9*b^5+20*a^9*b^4*c-26*a^9*b^3*c^2-26*a^9*b^2*c^3+20*a^9*b*c^4-3*a^9*c^5-3*a^8*b^6-9*a^8*b^5*c+70*a^8*b^4*c^2-12*a^8*b^3*c^3+70*a^8*b^2*c^4-9*a^8*b*c^5-3*a^8*c^6+2*a^7*b^7-2*a^7*b^6*c-37*a^7*b^5*c^2+77*a^7*b^4*c^3+77*a^7*b^3*c^4-37*a^7*b^2*c^5-2*a^7*b*c^6+2*a^7*c^7-3*a^6*b^8-2*a^6*b^7*c-50*a^6*b^6*c^2-37*a^6*b^5*c^3-120*a^6*b^4*c^4-37*a^6*b^3*c^5-50*a^6*b^2*c^6-2*a^6*b*c^7-3*a^6*c^8-3*a^5*b^9-9*a^5*b^8*c-37*a^5*b^7*c^2-37*a^5*b^6*c^3+62*a^5*b^5*c^4+62*a^5*b^4*c^5-37*a^5*b^3*c^6-37*a^5*b^2*c^7-9*a^5*b*c^8-3*a^5*c^9+6*a^4*b^10+20*a^4*b^9*c+70*a^4*b^8*c^2+77*a^4*b^7*c^3-120*a^4*b^6*c^4+62*a^4*b^5*c^5-120*a^4*b^4*c^6+77*a^4*b^3*c^7+70*a^4*b^2*c^8+20*a^4*b*c^9+6*a^4*c^10+3*a^3*b^11-5*a^3*b^10*c-26*a^3*b^9*c^2-12*a^3*b^8*c^3+77*a^3*b^7*c^4-37*a^3*b^6*c^5-37*a^3*b^5*c^6+77*a^3*b^4*c^7-12*a^3*b^3*c^8-26*a^3*b^2*c^9-5*a^3*b*c^10+3*a^3*c^11-4*a^2*b^12-9*a^2*b^11*c+31*a^2*b^10*c^2-26*a^2*b^9*c^3+70*a^2*b^8*c^4-37*a^2*b^7*c^5-50*a^2*b^6*c^6-37*a^2*b^5*c^7+70*a^2*b^4*c^8-26*a^2*b^3*c^9+31*a^2*b^2*c^10-9*a^2*b*c^11-4*a^2*c^12-a*b^13+6*a*b^12*c-9*a*b^11*c^2-5*a*b^10*c^3+20*a*b^9*c^4-9*a*b^8*c^5-2*a*b^7*c^6-2*a*b^6*c^7-9*a*b^5*c^8+20*a*b^4*c^9-5*a*b^3*c^10-9*a*b^2*c^11+6*a*b*c^12-a*c^13+b^14-b^13*c-4*b^12*c^2+3*b^11*c^3+6*b^10*c^4-3*b^9*c^5-3*b^8*c^6+2*b^7*c^7-3*b^6*c^8-3*b^5*c^9+6*b^4*c^10+3*b^3*c^11-4*b^2*c^12-b*c^13+c^14)*(a+b+c)^2*IF^4+(3*(a+b+c))*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(2*a^17-3*a^16*b-3*a^16*c-10*a^15*b^2+24*a^15*b*c-10*a^15*c^2+20*a^14*b^3-27*a^14*b^2*c-27*a^14*b*c^2+20*a^14*c^3+15*a^13*b^4-75*a^13*b^3*c+126*a^13*b^2*c^2-75*a^13*b*c^3+15*a^13*c^4-55*a^12*b^5+132*a^12*b^4*c-76*a^12*b^3*c^2-76*a^12*b^2*c^3+132*a^12*b*c^4-55*a^12*c^5+8*a^11*b^6+81*a^11*b^5*c-30*a^11*b^4*c^2+450*a^11*b^3*c^3-30*a^11*b^2*c^4+81*a^11*b*c^5+8*a^11*c^6+78*a^10*b^7-237*a^10*b^6*c+102*a^10*b^5*c^2-432*a^10*b^4*c^3-432*a^10*b^3*c^4+102*a^10*b^2*c^5-237*a^10*b*c^6+78*a^10*c^7-55*a^9*b^8-30*a^9*b^7*c-409*a^9*b^6*c^2+171*a^9*b^5*c^3-534*a^9*b^4*c^4+171*a^9*b^3*c^5-409*a^9*b^2*c^6-30*a^9*b*c^7-55*a^9*c^8-55*a^8*b^9+270*a^8*b^8*c+324*a^8*b^7*c^2-376*a^8*b^6*c^3+855*a^8*b^5*c^4+855*a^8*b^4*c^5-376*a^8*b^3*c^6+324*a^8*b^2*c^7+270*a^8*b*c^8-55*a^8*c^9+78*a^7*b^10-30*a^7*b^9*c+324*a^7*b^8*c^2+636*a^7*b^7*c^3-6*a^7*b^6*c^4-1404*a^7*b^5*c^5-6*a^7*b^4*c^6+636*a^7*b^3*c^7+324*a^7*b^2*c^8-30*a^7*b*c^9+78*a^7*c^10+8*a^6*b^11-237*a^6*b^10*c-409*a^6*b^9*c^2-376*a^6*b^8*c^3-6*a^6*b^7*c^4+500*a^6*b^6*c^5+500*a^6*b^5*c^6-6*a^6*b^4*c^7-376*a^6*b^3*c^8-409*a^6*b^2*c^9-237*a^6*b*c^10+8*a^6*c^11-55*a^5*b^12+81*a^5*b^11*c+102*a^5*b^10*c^2+171*a^5*b^9*c^3+855*a^5*b^8*c^4-1404*a^5*b^7*c^5+500*a^5*b^6*c^6-1404*a^5*b^5*c^7+855*a^5*b^4*c^8+171*a^5*b^3*c^9+102*a^5*b^2*c^10+81*a^5*b*c^11-55*a^5*c^12+15*a^4*b^13+132*a^4*b^12*c-30*a^4*b^11*c^2-432*a^4*b^10*c^3-534*a^4*b^9*c^4+855*a^4*b^8*c^5-6*a^4*b^7*c^6-6*a^4*b^6*c^7+855*a^4*b^5*c^8-534*a^4*b^4*c^9-432*a^4*b^3*c^10-30*a^4*b^2*c^11+132*a^4*b*c^12+15*a^4*c^13+20*a^3*b^14-75*a^3*b^13*c-76*a^3*b^12*c^2+450*a^3*b^11*c^3-432*a^3*b^10*c^4+171*a^3*b^9*c^5-376*a^3*b^8*c^6+636*a^3*b^7*c^7-376*a^3*b^6*c^8+171*a^3*b^5*c^9-432*a^3*b^4*c^10+450*a^3*b^3*c^11-76*a^3*b^2*c^12-75*a^3*b*c^13+20*a^3*c^14-10*a^2*b^15-27*a^2*b^14*c+126*a^2*b^13*c^2-76*a^2*b^12*c^3-30*a^2*b^11*c^4+102*a^2*b^10*c^5-409*a^2*b^9*c^6+324*a^2*b^8*c^7+324*a^2*b^7*c^8-409*a^2*b^6*c^9+102*a^2*b^5*c^10-30*a^2*b^4*c^11-76*a^2*b^3*c^12+126*a^2*b^2*c^13-27*a^2*b*c^14-10*a^2*c^15-3*a*b^16+24*a*b^15*c-27*a*b^14*c^2-75*a*b^13*c^3+132*a*b^12*c^4+81*a*b^11*c^5-237*a*b^10*c^6-30*a*b^9*c^7+270*a*b^8*c^8-30*a*b^7*c^9-237*a*b^6*c^10+81*a*b^5*c^11+132*a*b^4*c^12-75*a*b^3*c^13-27*a*b^2*c^14+24*a*b*c^15-3*a*c^16+2*b^17-3*b^16*c-10*b^15*c^2+20*b^14*c^3+15*b^13*c^4-55*b^12*c^5+8*b^11*c^6+78*b^10*c^7-55*b^9*c^8-55*b^8*c^9+78*b^7*c^10+8*b^6*c^11-55*b^5*c^12+15*b^4*c^13+20*b^3*c^14-10*b^2*c^15-3*b*c^16+2*c^17)*IF^2+(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(a^10-a^9*b-a^9*c-3*a^8*b^2+5*a^8*b*c-3*a^8*c^2+4*a^7*b^3-2*a^7*b^2*c-2*a^7*b*c^2+4*a^7*c^3+2*a^6*b^4-14*a^6*b^3*c-4*a^6*b^2*c^2-14*a^6*b*c^3+2*a^6*c^4-6*a^5*b^5+12*a^5*b^4*c-34*a^5*b^3*c^2-34*a^5*b^2*c^3+12*a^5*b*c^4-6*a^5*c^5+2*a^4*b^6+12*a^4*b^5*c+86*a^4*b^4*c^2+44*a^4*b^3*c^3+86*a^4*b^2*c^4+12*a^4*b*c^5+2*a^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5.    81*(a^2+b^2+c^2)^2*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)^2*(a-b-c)^4*O1F^8+(27*(a^2+b^2+c^2))*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(2*a^9+a^8*b+a^8*c-6*a^7*b^2+12*a^7*b*c-6*a^7*c^2+18*a^6*b^2*c+18*a^6*b*c^2+5*a^5*b^4-16*a^5*b^2*c^2+5*a^5*c^4-5*a^4*b^5-14*a^4*b^4*c-20*a^4*b^3*c^2-20*a^4*b^2*c^3-14*a^4*b*c^4-5*a^4*c^5+20*a^3*b^4*c^2-36*a^3*b^3*c^3+20*a^3*b^2*c^4+6*a^2*b^7+18*a^2*b^6*c+16*a^2*b^5*c^2-20*a^2*b^4*c^3-20*a^2*b^3*c^4+16*a^2*b^2*c^5+18*a^2*b*c^6+6*a^2*c^7-a*b^8+12*a*b^7*c-18*a*b^6*c^2+14*a*b^4*c^4-18*a*b^2*c^6+12*a*b*c^7-a*c^8-2*b^9+b^8*c+6*b^7*c^2-5*b^5*c^4-5*b^4*c^5+6*b^2*c^7+b*c^8-2*c^9)*(a-b-c)^3*O_1F^6+(27*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a^14+a^13*b+a^13*c-4*a^12*b^2+6*a^12*b*c-4*a^12*c^2-3*a^11*b^3+9*a^11*b^2*c+9*a^11*b*c^2-3*a^11*c^3+6*a^10*b^4-5*a^10*b^3*c+31*a^10*b^2*c^2-5*a^10*b*c^3+6*a^10*c^4+3*a^9*b^5-20*a^9*b^4*c+26*a^9*b^3*c^2+26*a^9*b^2*c^3-20*a^9*b*c^4+3*a^9*c^5-3*a^8*b^6-9*a^8*b^5*c+70*a^8*b^4*c^2-12*a^8*b^3*c^3+70*a^8*b^2*c^4-9*a^8*b*c^5-3*a^8*c^6-2*a^7*b^7+2*a^7*b^6*c+37*a^7*b^5*c^2-77*a^7*b^4*c^3-77*a^7*b^3*c^4+37*a^7*b^2*c^5+2*a^7*b*c^6-2*a^7*c^7-3*a^6*b^8-2*a^6*b^7*c-50*a^6*b^6*c^2-37*a^6*b^5*c^3-120*a^6*b^4*c^4-37*a^6*b^3*c^5-50*a^6*b^2*c^6-2*a^6*b*c^7-3*a^6*c^8+3*a^5*b^9+9*a^5*b^8*c+37*a^5*b^7*c^2+37*a^5*b^6*c^3-62*a^5*b^5*c^4-62*a^5*b^4*c^5+37*a^5*b^3*c^6+37*a^5*b^2*c^7+9*a^5*b*c^8+3*a^5*c^9+6*a^4*b^10+20*a^4*b^9*c+70*a^4*b^8*c^2+77*a^4*b^7*c^3-120*a^4*b^6*c^4+62*a^4*b^5*c^5-120*a^4*b^4*c^6+77*a^4*b^3*c^7+70*a^4*b^2*c^8+20*a^4*b*c^9+6*a^4*c^10-3*a^3*b^11+5*a^3*b^10*c+26*a^3*b^9*c^2+12*a^3*b^8*c^3-77*a^3*b^7*c^4+37*a^3*b^6*c^5+37*a^3*b^5*c^6-77*a^3*b^4*c^7+12*a^3*b^3*c^8+26*a^3*b^2*c^9+5*a^3*b*c^10-3*a^3*c^11-4*a^2*b^12-9*a^2*b^11*c+31*a^2*b^10*c^2-26*a^2*b^9*c^3+70*a^2*b^8*c^4-37*a^2*b^7*c^5-50*a^2*b^6*c^6-37*a^2*b^5*c^7+70*a^2*b^4*c^8-26*a^2*b^3*c^9+31*a^2*b^2*c^10-9*a^2*b*c^11-4*a^2*c^12+a*b^13-6*a*b^12*c+9*a*b^11*c^2+5*a*b^10*c^3-20*a*b^9*c^4+9*a*b^8*c^5+2*a*b^7*c^6+2*a*b^6*c^7+9*a*b^5*c^8-20*a*b^4*c^9+5*a*b^3*c^10+9*a*b^2*c^11-6*a*b*c^12+a*c^13+b^14-b^13*c-4*b^12*c^2+3*b^11*c^3+6*b^10*c^4-3*b^9*c^5-3*b^8*c^6+2*b^7*c^7-3*b^6*c^8-3*b^5*c^9+6*b^4*c^10+3*b^3*c^11-4*b^2*c^12-b*c^13+c^14)*(a-b-c)^2*O1F^4+(3*(a-b-c))*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(2*a^17+3*a^16*b+3*a^16*c-10*a^15*b^2+24*a^15*b*c-10*a^15*c^2-20*a^14*b^3+27*a^14*b^2*c+27*a^14*b*c^2-20*a^14*c^3+15*a^13*b^4-75*a^13*b^3*c+126*a^13*b^2*c^2-75*a^13*b*c^3+15*a^13*c^4+55*a^12*b^5-132*a^12*b^4*c+76*a^12*b^3*c^2+76*a^12*b^2*c^3-132*a^12*b*c^4+55*a^12*c^5+8*a^11*b^6+81*a^11*b^5*c-30*a^11*b^4*c^2+450*a^11*b^3*c^3-30*a^11*b^2*c^4+81*a^11*b*c^5+8*a^11*c^6-78*a^10*b^7+237*a^10*b^6*c-102*a^10*b^5*c^2+432*a^10*b^4*c^3+432*a^10*b^3*c^4-102*a^10*b^2*c^5+237*a^10*b*c^6-78*a^10*c^7-55*a^9*b^8-30*a^9*b^7*c-409*a^9*b^6*c^2+171*a^9*b^5*c^3-534*a^9*b^4*c^4+171*a^9*b^3*c^5-409*a^9*b^2*c^6-30*a^9*b*c^7-55*a^9*c^8+55*a^8*b^9-270*a^8*b^8*c-324*a^8*b^7*c^2+376*a^8*b^6*c^3-855*a^8*b^5*c^4-855*a^8*b^4*c^5+376*a^8*b^3*c^6-324*a^8*b^2*c^7-270*a^8*b*c^8+55*a^8*c^9+78*a^7*b^10-30*a^7*b^9*c+324*a^7*b^8*c^2+636*a^7*b^7*c^3-6*a^7*b^6*c^4-1404*a^7*b^5*c^5-6*a^7*b^4*c^6+636*a^7*b^3*c^7+324*a^7*b^2*c^8-30*a^7*b*c^9+78*a^7*c^10-8*a^6*b^11+237*a^6*b^10*c+409*a^6*b^9*c^2+376*a^6*b^8*c^3+6*a^6*b^7*c^4-500*a^6*b^6*c^5-500*a^6*b^5*c^6+6*a^6*b^4*c^7+376*a^6*b^3*c^8+409*a^6*b^2*c^9+237*a^6*b*c^10-8*a^6*c^11-55*a^5*b^12+81*a^5*b^11*c+102*a^5*b^10*c^2+171*a^5*b^9*c^3+855*a^5*b^8*c^4-1404*a^5*b^7*c^5+500*a^5*b^6*c^6-1404*a^5*b^5*c^7+855*a^5*b^4*c^8+171*a^5*b^3*c^9+102*a^5*b^2*c^10+81*a^5*b*c^11-55*a^5*c^12-15*a^4*b^13-132*a^4*b^12*c+30*a^4*b^11*c^2+432*a^4*b^10*c^3+534*a^4*b^9*c^4-855*a^4*b^8*c^5+6*a^4*b^7*c^6+6*a^4*b^6*c^7-855*a^4*b^5*c^8+534*a^4*b^4*c^9+432*a^4*b^3*c^10+30*a^4*b^2*c^11-132*a^4*b*c^12-15*a^4*c^13+20*a^3*b^14-75*a^3*b^13*c-76*a^3*b^12*c^2+450*a^3*b^11*c^3-432*a^3*b^10*c^4+171*a^3*b^9*c^5-376*a^3*b^8*c^6+636*a^3*b^7*c^7-376*a^3*b^6*c^8+171*a^3*b^5*c^9-432*a^3*b^4*c^10+450*a^3*b^3*c^11-76*a^3*b^2*c^12-75*a^3*b*c^13+20*a^3*c^14+10*a^2*b^15+27*a^2*b^14*c-126*a^2*b^13*c^2+76*a^2*b^12*c^3+30*a^2*b^11*c^4-102*a^2*b^10*c^5+409*a^2*b^9*c^6-324*a^2*b^8*c^7-324*a^2*b^7*c^8+409*a^2*b^6*c^9-102*a^2*b^5*c^10+30*a^2*b^4*c^11+76*a^2*b^3*c^12-126*a^2*b^2*c^13+27*a^2*b*c^14+10*a^2*c^15-3*a*b^16+24*a*b^15*c-27*a*b^14*c^2-75*a*b^13*c^3+132*a*b^12*c^4+81*a*b^11*c^5-237*a*b^10*c^6-30*a*b^9*c^7+270*a*b^8*c^8-30*a*b^7*c^9-237*a*b^6*c^10+81*a*b^5*c^11+132*a*b^4*c^12-75*a*b^3*c^13-27*a*b^2*c^14+24*a*b*c^15-3*a*c^16-2*b^17+3*b^16*c+10*b^15*c^2-20*b^14*c^3-15*b^13*c^4+55*b^12*c^5-8*b^11*c^6-78*b^10*c^7+55*b^9*c^8+55*b^8*c^9-78*b^7*c^10-8*b^6*c^11+55*b^5*c^12-15*b^4*c^13-20*b^3*c^14+10*b^2*c^15+3*b*c^16-2*c^17)*O1F^2+(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(a^10+a^9*b+a^9*c-3*a^8*b^2+5*a^8*b*c-3*a^8*c^2-4*a^7*b^3+2*a^7*b^2*c+2*a^7*b*c^2-4*a^7*c^3+2*a^6*b^4-14*a^6*b^3*c+23*a^6*b^2*c^2-14*a^6*b*c^3+2*a^6*c^4+6*a^5*b^5-12*a^5*b^4*c+7*a^5*b^3*c^2+7*a^5*b^2*c^3-12*a^5*b*c^4+6*a^5*c^5+2*a^4*b^6+12*a^4*b^5*c-22*a^4*b^4*c^2+17*a^4*b^3*c^3-22*a^4*b^2*c^4+12*a^4*b*c^5+2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6.   81*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)^2*(a-b+c)^4*(a^2+b^2+c^2)^2*O2F^8+(27*(a^2+b^2+c^2))*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(2*a^9+a^8*b-a^8*c-6*a^7*b^2-12*a^7*b*c-6*a^7*c^2-18*a^6*b^2*c+18*a^6*b*c^2+5*a^5*b^4-16*a^5*b^2*c^2+5*a^5*c^4-5*a^4*b^5+14*a^4*b^4*c-20*a^4*b^3*c^2+20*a^4*b^2*c^3-14*a^4*b*c^4+5*a^4*c^5+20*a^3*b^4*c^2+36*a^3*b^3*c^3+20*a^3*b^2*c^4+6*a^2*b^7-18*a^2*b^6*c+16*a^2*b^5*c^2+20*a^2*b^4*c^3-20*a^2*b^3*c^4-16*a^2*b^2*c^5+18*a^2*b*c^6-6*a^2*c^7-a*b^8-12*a*b^7*c-18*a*b^6*c^2+14*a*b^4*c^4-18*a*b^2*c^6-12*a*b*c^7-a*c^8-2*b^9-b^8*c+6*b^7*c^2-5*b^5*c^4+5*b^4*c^5-6*b^2*c^7+b*c^8+2*c^9)*(a-b+c)^3*O2F^6+(27*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a^14+a^13*b-a^13*c-4*a^12*b^2-6*a^12*b*c-4*a^12*c^2-3*a^11*b^3-9*a^11*b^2*c+9*a^11*b*c^2+3*a^11*c^3+6*a^10*b^4+5*a^10*b^3*c+31*a^10*b^2*c^2+5*a^10*b*c^3+6*a^10*c^4+3*a^9*b^5+20*a^9*b^4*c+26*a^9*b^3*c^2-26*a^9*b^2*c^3-20*a^9*b*c^4-3*a^9*c^5-3*a^8*b^6+9*a^8*b^5*c+70*a^8*b^4*c^2+12*a^8*b^3*c^3+70*a^8*b^2*c^4+9*a^8*b*c^5-3*a^8*c^6-2*a^7*b^7-2*a^7*b^6*c+37*a^7*b^5*c^2+77*a^7*b^4*c^3-77*a^7*b^3*c^4-37*a^7*b^2*c^5+2*a^7*b*c^6+2*a^7*c^7-3*a^6*b^8+2*a^6*b^7*c-50*a^6*b^6*c^2+37*a^6*b^5*c^3-120*a^6*b^4*c^4+37*a^6*b^3*c^5-50*a^6*b^2*c^6+2*a^6*b*c^7-3*a^6*c^8+3*a^5*b^9-9*a^5*b^8*c+37*a^5*b^7*c^2-37*a^5*b^6*c^3-62*a^5*b^5*c^4+62*a^5*b^4*c^5+37*a^5*b^3*c^6-37*a^5*b^2*c^7+9*a^5*b*c^8-3*a^5*c^9+6*a^4*b^10-20*a^4*b^9*c+70*a^4*b^8*c^2-77*a^4*b^7*c^3-120*a^4*b^6*c^4-62*a^4*b^5*c^5-120*a^4*b^4*c^6-77*a^4*b^3*c^7+70*a^4*b^2*c^8-20*a^4*b*c^9+6*a^4*c^10-3*a^3*b^11-5*a^3*b^10*c+26*a^3*b^9*c^2-12*a^3*b^8*c^3-77*a^3*b^7*c^4-37*a^3*b^6*c^5+37*a^3*b^5*c^6+77*a^3*b^4*c^7+12*a^3*b^3*c^8-26*a^3*b^2*c^9+5*a^3*b*c^10+3*a^3*c^11-4*a^2*b^12+9*a^2*b^11*c+31*a^2*b^10*c^2+26*a^2*b^9*c^3+70*a^2*b^8*c^4+37*a^2*b^7*c^5-50*a^2*b^6*c^6+37*a^2*b^5*c^7+70*a^2*b^4*c^8+26*a^2*b^3*c^9+31*a^2*b^2*c^10+9*a^2*b*c^11-4*a^2*c^12+a*b^13+6*a*b^12*c+9*a*b^11*c^2-5*a*b^10*c^3-20*a*b^9*c^4-9*a*b^8*c^5+2*a*b^7*c^6-2*a*b^6*c^7+9*a*b^5*c^8+20*a*b^4*c^9+5*a*b^3*c^10-9*a*b^2*c^11-6*a*b*c^12-a*c^13+b^14+b^13*c-4*b^12*c^2-3*b^11*c^3+6*b^10*c^4+3*b^9*c^5-3*b^8*c^6-2*b^7*c^7-3*b^6*c^8+3*b^5*c^9+6*b^4*c^10-3*b^3*c^11-4*b^2*c^12+b*c^13+c^14)*(a-b+c)^2*O2F^4+(3*(a-b+c))*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(2*a^17+3*a^16*b-3*a^16*c-10*a^15*b^2-24*a^15*b*c-10*a^15*c^2-20*a^14*b^3-27*a^14*b^2*c+27*a^14*b*c^2+20*a^14*c^3+15*a^13*b^4+75*a^13*b^3*c+126*a^13*b^2*c^2+75*a^13*b*c^3+15*a^13*c^4+55*a^12*b^5+132*a^12*b^4*c+76*a^12*b^3*c^2-76*a^12*b^2*c^3-132*a^12*b*c^4-55*a^12*c^5+8*a^11*b^6-81*a^11*b^5*c-30*a^11*b^4*c^2-450*a^11*b^3*c^3-30*a^11*b^2*c^4-81*a^11*b*c^5+8*a^11*c^6-78*a^10*b^7-237*a^10*b^6*c-102*a^10*b^5*c^2-432*a^10*b^4*c^3+432*a^10*b^3*c^4+102*a^10*b^2*c^5+237*a^10*b*c^6+78*a^10*c^7-55*a^9*b^8+30*a^9*b^7*c-409*a^9*b^6*c^2-171*a^9*b^5*c^3-534*a^9*b^4*c^4-171*a^9*b^3*c^5-409*a^9*b^2*c^6+30*a^9*b*c^7-55*a^9*c^8+55*a^8*b^9+270*a^8*b^8*c-324*a^8*b^7*c^2-376*a^8*b^6*c^3-855*a^8*b^5*c^4+855*a^8*b^4*c^5+376*a^8*b^3*c^6+324*a^8*b^2*c^7-270*a^8*b*c^8-55*a^8*c^9+78*a^7*b^10+30*a^7*b^9*c+324*a^7*b^8*c^2-636*a^7*b^7*c^3-6*a^7*b^6*c^4+1404*a^7*b^5*c^5-6*a^7*b^4*c^6-636*a^7*b^3*c^7+324*a^7*b^2*c^8+30*a^7*b*c^9+78*a^7*c^10-8*a^6*b^11-237*a^6*b^10*c+409*a^6*b^9*c^2-376*a^6*b^8*c^3+6*a^6*b^7*c^4+500*a^6*b^6*c^5-500*a^6*b^5*c^6-6*a^6*b^4*c^7+376*a^6*b^3*c^8-409*a^6*b^2*c^9+237*a^6*b*c^10+8*a^6*c^11-55*a^5*b^12-81*a^5*b^11*c+102*a^5*b^10*c^2-171*a^5*b^9*c^3+855*a^5*b^8*c^4+1404*a^5*b^7*c^5+500*a^5*b^6*c^6+1404*a^5*b^5*c^7+855*a^5*b^4*c^8-171*a^5*b^3*c^9+102*a^5*b^2*c^10-81*a^5*b*c^11-55*a^5*c^12-15*a^4*b^13+132*a^4*b^12*c+30*a^4*b^11*c^2-432*a^4*b^10*c^3+534*a^4*b^9*c^4+855*a^4*b^8*c^5+6*a^4*b^7*c^6-6*a^4*b^6*c^7-855*a^4*b^5*c^8-534*a^4*b^4*c^9+432*a^4*b^3*c^10-30*a^4*b^2*c^11-132*a^4*b*c^12+15*a^4*c^13+20*a^3*b^14+75*a^3*b^13*c-76*a^3*b^12*c^2-450*a^3*b^11*c^3-432*a^3*b^10*c^4-171*a^3*b^9*c^5-376*a^3*b^8*c^6-636*a^3*b^7*c^7-376*a^3*b^6*c^8-171*a^3*b^5*c^9-432*a^3*b^4*c^10-450*a^3*b^3*c^11-76*a^3*b^2*c^12+75*a^3*b*c^13+20*a^3*c^14+10*a^2*b^15-27*a^2*b^14*c-126*a^2*b^13*c^2-76*a^2*b^12*c^3+30*a^2*b^11*c^4+102*a^2*b^10*c^5+409*a^2*b^9*c^6+324*a^2*b^8*c^7-324*a^2*b^7*c^8-409*a^2*b^6*c^9-102*a^2*b^5*c^10-30*a^2*b^4*c^11+76*a^2*b^3*c^12+126*a^2*b^2*c^13+27*a^2*b*c^14-10*a^2*c^15-3*a*b^16-24*a*b^15*c-27*a*b^14*c^2+75*a*b^13*c^3+132*a*b^12*c^4-81*a*b^11*c^5-237*a*b^10*c^6+30*a*b^9*c^7+270*a*b^8*c^8+30*a*b^7*c^9-237*a*b^6*c^10-81*a*b^5*c^11+132*a*b^4*c^12+75*a*b^3*c^13-27*a*b^2*c^14-24*a*b*c^15-3*a*c^16-2*b^17-3*b^16*c+10*b^15*c^2+20*b^14*c^3-15*b^13*c^4-55*b^12*c^5-8*b^11*c^6+78*b^10*c^7+55*b^9*c^8-55*b^8*c^9-78*b^7*c^10+8*b^6*c^11+55*b^5*c^12+15*b^4*c^13-20*b^3*c^14-10*b^2*c^15+3*b*c^16+2*c^17)*O2F^2+(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(a^10+a^9*b-a^9*c-3*a^8*b^2-5*a^8*b*c-3*a^8*c^2-4*a^7*b^3-2*a^7*b^2*c+2*a^7*b*c^2+4*a^7*c^3+2*a^6*b^4+14*a^6*b^3*c-4*a^6*b^2*c^2+14*a^6*b*c^3+2*a^6*c^4+6*a^5*b^5+12*a^5*b^4*c+34*a^5*b^3*c^2-34*a^5*b^2*c^3-12*a^5*b*c^4-6*a^5*c^5+2*a^4*b^6-12*a^4*b^5*c+86*a^4*b^4*c^2-44*a^4*b^3*c^3+86*a^4*b^2*c^4-12*a^4*b*c^5+2*a^4*c^6-4*a^3*b^7-14*a^3*b^6*c+34*a^3*b^5*c^2+44*a^3*b^4*c^3-44*a^3*b^3*c^4-34*a^3*b^2*c^5+14*a^3*b*c^6+4*a^3*c^7-3*a^2*b^8+2*a^2*b^7*c-4*a^2*b^6*c^2+34*a^2*b^5*c^3+86*a^2*b^4*c^4+34*a^2*b^3*c^5-4*a^2*b^2*c^6+2*a^2*b*c^7-3*a^2*c^8+a*b^9+5*a*b^8*c+2*a*b^7*c^2-14*a*b^6*c^3-12*a*b^5*c^4+12*a*b^4*c^5+14*a*b^3*c^6-2*a*b^2*c^7-5*a*b*c^8-a*c^9+b^10+b^9*c-3*b^8*c^2-4*b^7*c^3+2*b^6*c^4+6*b^5*c^5+2*b^4*c^6-4*b^3*c^7-3*b^2*c^8+b*c^9+c^10)*(a^10+a^9*b-a^9*c-3*a^8*b^2-5*a^8*b*c-3*a^8*c^2-4*a^7*b^3-2*a^7*b^2*c+2*a^7*b*c^2+4*a^7*c^3+2*a^6*b^4+14*a^6*b^3*c+23*a^6*b^2*c^2+14*a^6*b*c^3+2*a^6*c^4+6*a^5*b^5+12*a^5*b^4*c+7*a^5*b^3*c^2-7*a^5*b^2*c^3-12*a^5*b*c^4-6*a^5*c^5+2*a^4*b^6-12*a^4*b^5*c-22*a^4*b^4*c^2-17*a^4*b^3*c^3-22*a^4*b^2*c^4-12*a^4*b*c^5+2*a^4*c^6-4*a^3*b^7-14*a^3*b^6*c+7*a^3*b^5*c^2+17*a^3*b^4*c^3-17*a^3*b^3*c^4-7*a^3*b^2*c^5+14*a^3*b*c^6+4*a^3*c^7-3*a^2*b^8+2*a^2*b^7*c+23*a^2*b^6*c^2+7*a^2*b^5*c^3-22*a^2*b^4*c^4+7*a^2*b^3*c^5+23*a^2*b^2*c^6+2*a^2*b*c^7-3*a^2*c^8+a*b^9+5*a*b^8*c+2*a*b^7*c^2-14*a*b^6*c^3-12*a*b^5*c^4+12*a*b^4*c^5+14*a*b^3*c^6-2*a*b^2*c^7-5*a*b*c^8-a*c^9+b^10+b^9*c-3*b^8*c^2-4*b^7*c^3+2*b^6*c^4+6*b^5*c^5+2*b^4*c^6-4*b^3*c^7-3*b^2*c^8+b*c^9+c^10)=0


7.   81*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)^2*(a+b-c)^4*(a^2+b^2+c^2)^2*O3F^8+(27*(a^2+b^2+c^2))*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(2*a^9-a^8*b+a^8*c-6*a^7*b^2-12*a^7*b*c-6*a^7*c^2+18*a^6*b^2*c-18*a^6*b*c^2+5*a^5*b^4-16*a^5*b^2*c^2+5*a^5*c^4+5*a^4*b^5-14*a^4*b^4*c+20*a^4*b^3*c^2-20*a^4*b^2*c^3+14*a^4*b*c^4-5*a^4*c^5+20*a^3*b^4*c^2+36*a^3*b^3*c^3+20*a^3*b^2*c^4-6*a^2*b^7+18*a^2*b^6*c-16*a^2*b^5*c^2-20*a^2*b^4*c^3+20*a^2*b^3*c^4+16*a^2*b^2*c^5-18*a^2*b*c^6+6*a^2*c^7-a*b^8-12*a*b^7*c-18*a*b^6*c^2+14*a*b^4*c^4-18*a*b^2*c^6-12*a*b*c^7-a*c^8+2*b^9+b^8*c-6*b^7*c^2+5*b^5*c^4-5*b^4*c^5+6*b^2*c^7-b*c^8-2*c^9)*(a+b-c)^3*O3F^6+(27*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a^14-a^13*b+a^13*c-4*a^12*b^2-6*a^12*b*c-4*a^12*c^2+3*a^11*b^3+9*a^11*b^2*c-9*a^11*b*c^2-3*a^11*c^3+6*a^10*b^4+5*a^10*b^3*c+31*a^10*b^2*c^2+5*a^10*b*c^3+6*a^10*c^4-3*a^9*b^5-20*a^9*b^4*c-26*a^9*b^3*c^2+26*a^9*b^2*c^3+20*a^9*b*c^4+3*a^9*c^5-3*a^8*b^6+9*a^8*b^5*c+70*a^8*b^4*c^2+12*a^8*b^3*c^3+70*a^8*b^2*c^4+9*a^8*b*c^5-3*a^8*c^6+2*a^7*b^7+2*a^7*b^6*c-37*a^7*b^5*c^2-77*a^7*b^4*c^3+77*a^7*b^3*c^4+37*a^7*b^2*c^5-2*a^7*b*c^6-2*a^7*c^7-3*a^6*b^8+2*a^6*b^7*c-50*a^6*b^6*c^2+37*a^6*b^5*c^3-120*a^6*b^4*c^4+37*a^6*b^3*c^5-50*a^6*b^2*c^6+2*a^6*b*c^7-3*a^6*c^8-3*a^5*b^9+9*a^5*b^8*c-37*a^5*b^7*c^2+37*a^5*b^6*c^3+62*a^5*b^5*c^4-62*a^5*b^4*c^5-37*a^5*b^3*c^6+37*a^5*b^2*c^7-9*a^5*b*c^8+3*a^5*c^9+6*a^4*b^10-20*a^4*b^9*c+70*a^4*b^8*c^2-77*a^4*b^7*c^3-120*a^4*b^6*c^4-62*a^4*b^5*c^5-120*a^4*b^4*c^6-77*a^4*b^3*c^7+70*a^4*b^2*c^8-20*a^4*b*c^9+6*a^4*c^10+3*a^3*b^11+5*a^3*b^10*c-26*a^3*b^9*c^2+12*a^3*b^8*c^3+77*a^3*b^7*c^4+37*a^3*b^6*c^5-37*a^3*b^5*c^6-77*a^3*b^4*c^7-12*a^3*b^3*c^8+26*a^3*b^2*c^9-5*a^3*b*c^10-3*a^3*c^11-4*a^2*b^12+9*a^2*b^11*c+31*a^2*b^10*c^2+26*a^2*b^9*c^3+70*a^2*b^8*c^4+37*a^2*b^7*c^5-50*a^2*b^6*c^6+37*a^2*b^5*c^7+70*a^2*b^4*c^8+26*a^2*b^3*c^9+31*a^2*b^2*c^10+9*a^2*b*c^11-4*a^2*c^12-a*b^13-6*a*b^12*c-9*a*b^11*c^2+5*a*b^10*c^3+20*a*b^9*c^4+9*a*b^8*c^5-2*a*b^7*c^6+2*a*b^6*c^7-9*a*b^5*c^8-20*a*b^4*c^9-5*a*b^3*c^10+9*a*b^2*c^11+6*a*b*c^12+a*c^13+b^14+b^13*c-4*b^12*c^2-3*b^11*c^3+6*b^10*c^4+3*b^9*c^5-3*b^8*c^6-2*b^7*c^7-3*b^6*c^8+3*b^5*c^9+6*b^4*c^10-3*b^3*c^11-4*b^2*c^12+b*c^13+c^14)*(a+b-c)^2*O3F^4+(3*(a+b-c))*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(2*a^17-3*a^16*b+3*a^16*c-10*a^15*b^2-24*a^15*b*c-10*a^15*c^2+20*a^14*b^3+27*a^14*b^2*c-27*a^14*b*c^2-20*a^14*c^3+15*a^13*b^4+75*a^13*b^3*c+126*a^13*b^2*c^2+75*a^13*b*c^3+15*a^13*c^4-55*a^12*b^5-132*a^12*b^4*c-76*a^12*b^3*c^2+76*a^12*b^2*c^3+132*a^12*b*c^4+55*a^12*c^5+8*a^11*b^6-81*a^11*b^5*c-30*a^11*b^4*c^2-450*a^11*b^3*c^3-30*a^11*b^2*c^4-81*a^11*b*c^5+8*a^11*c^6+78*a^10*b^7+237*a^10*b^6*c+102*a^10*b^5*c^2+432*a^10*b^4*c^3-432*a^10*b^3*c^4-102*a^10*b^2*c^5-237*a^10*b*c^6-78*a^10*c^7-55*a^9*b^8+30*a^9*b^7*c-409*a^9*b^6*c^2-171*a^9*b^5*c^3-534*a^9*b^4*c^4-171*a^9*b^3*c^5-409*a^9*b^2*c^6+30*a^9*b*c^7-55*a^9*c^8-55*a^8*b^9-270*a^8*b^8*c+324*a^8*b^7*c^2+376*a^8*b^6*c^3+855*a^8*b^5*c^4-855*a^8*b^4*c^5-376*a^8*b^3*c^6-324*a^8*b^2*c^7+270*a^8*b*c^8+55*a^8*c^9+78*a^7*b^10+30*a^7*b^9*c+324*a^7*b^8*c^2-636*a^7*b^7*c^3-6*a^7*b^6*c^4+1404*a^7*b^5*c^5-6*a^7*b^4*c^6-636*a^7*b^3*c^7+324*a^7*b^2*c^8+30*a^7*b*c^9+78*a^7*c^10+8*a^6*b^11+237*a^6*b^10*c-409*a^6*b^9*c^2+376*a^6*b^8*c^3-6*a^6*b^7*c^4-500*a^6*b^6*c^5+500*a^6*b^5*c^6+6*a^6*b^4*c^7-376*a^6*b^3*c^8+409*a^6*b^2*c^9-237*a^6*b*c^10-8*a^6*c^11-55*a^5*b^12-81*a^5*b^11*c+102*a^5*b^10*c^2-171*a^5*b^9*c^3+855*a^5*b^8*c^4+1404*a^5*b^7*c^5+500*a^5*b^6*c^6+1404*a^5*b^5*c^7+855*a^5*b^4*c^8-171*a^5*b^3*c^9+102*a^5*b^2*c^10-81*a^5*b*c^11-55*a^5*c^12+15*a^4*b^13-132*a^4*b^12*c-30*a^4*b^11*c^2+432*a^4*b^10*c^3-534*a^4*b^9*c^4-855*a^4*b^8*c^5-6*a^4*b^7*c^6+6*a^4*b^6*c^7+855*a^4*b^5*c^8+534*a^4*b^4*c^9-432*a^4*b^3*c^10+30*a^4*b^2*c^11+132*a^4*b*c^12-15*a^4*c^13+20*a^3*b^14+75*a^3*b^13*c-76*a^3*b^12*c^2-450*a^3*b^11*c^3-432*a^3*b^10*c^4-171*a^3*b^9*c^5-376*a^3*b^8*c^6-636*a^3*b^7*c^7-376*a^3*b^6*c^8-171*a^3*b^5*c^9-432*a^3*b^4*c^10-450*a^3*b^3*c^11-76*a^3*b^2*c^12+75*a^3*b*c^13+20*a^3*c^14-10*a^2*b^15+27*a^2*b^14*c+126*a^2*b^13*c^2+76*a^2*b^12*c^3-30*a^2*b^11*c^4-102*a^2*b^10*c^5-409*a^2*b^9*c^6-324*a^2*b^8*c^7+324*a^2*b^7*c^8+409*a^2*b^6*c^9+102*a^2*b^5*c^10+30*a^2*b^4*c^11-76*a^2*b^3*c^12-126*a^2*b^2*c^13-27*a^2*b*c^14+10*a^2*c^15-3*a*b^16-24*a*b^15*c-27*a*b^14*c^2+75*a*b^13*c^3+132*a*b^12*c^4-81*a*b^11*c^5-237*a*b^10*c^6+30*a*b^9*c^7+270*a*b^8*c^8+30*a*b^7*c^9-237*a*b^6*c^10-81*a*b^5*c^11+132*a*b^4*c^12+75*a*b^3*c^13-27*a*b^2*c^14-24*a*b*c^15-3*a*c^16+2*b^17+3*b^16*c-10*b^15*c^2-20*b^14*c^3+15*b^13*c^4+55*b^12*c^5+8*b^11*c^6-78*b^10*c^7-55*b^9*c^8+55*b^8*c^9+78*b^7*c^10-8*b^6*c^11-55*b^5*c^12-15*b^4*c^13+20*b^3*c^14+10*b^2*c^15-3*b*c^16-2*c^17)*O3F^2+(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(a^10-a^9*b+a^9*c-3*a^8*b^2-5*a^8*b*c-3*a^8*c^2+4*a^7*b^3+2*a^7*b^2*c-2*a^7*b*c^2-4*a^7*c^3+2*a^6*b^4+14*a^6*b^3*c+23*a^6*b^2*c^2+14*a^6*b*c^3+2*a^6*c^4-6*a^5*b^5-12*a^5*b^4*c-7*a^5*b^3*c^2+7*a^5*b^2*c^3+12*a^5*b*c^4+6*a^5*c^5+2*a^4*b^6-12*a^4*b^5*c-22*a^4*b^4*c^2-17*a^4*b^3*c^3-22*a^4*b^2*c^4-12*a^4*b*c^5+2*a^4*c^6+4*a^3*b^7+14*a^3*b^6*c-7*a^3*b^5*c^2-17*a^3*b^4*c^3+17*a^3*b^3*c^4+7*a^3*b^2*c^5-14*a^3*b*c^6-4*a^3*c^7-3*a^2*b^8+2*a^2*b^7*c+23*a^2*b^6*c^2+7*a^2*b^5*c^3-22*a^2*b^4*c^4+7*a^2*b^3*c^5+23*a^2*b^2*c^6+2*a^2*b*c^7-3*a^2*c^8-a*b^9-5*a*b^8*c-2*a*b^7*c^2+14*a*b^6*c^3+12*a*b^5*c^4-12*a*b^4*c^5-14*a*b^3*c^6+2*a*b^2*c^7+5*a*b*c^8+a*c^9+b^10+b^9*c-3*b^8*c^2-4*b^7*c^3+2*b^6*c^4+6*b^5*c^5+2*b^4*c^6-4*b^3*c^7-3*b^2*c^8+b*c^9+c^10)*(a^10-a^9*b+a^9*c-3*a^8*b^2-5*a^8*b*c-3*a^8*c^2+4*a^7*b^3+2*a^7*b^2*c-2*a^7*b*c^2-4*a^7*c^3+2*a^6*b^4+14*a^6*b^3*c-4*a^6*b^2*c^2+14*a^6*b*c^3+2*a^6*c^4-6*a^5*b^5-12*a^5*b^4*c-34*a^5*b^3*c^2+34*a^5*b^2*c^3+12*a^5*b*c^4+6*a^5*c^5+2*a^4*b^6-12*a^4*b^5*c+86*a^4*b^4*c^2-44*a^4*b^3*c^3+86*a^4*b^2*c^4-12*a^4*b*c^5+2*a^4*c^6+4*a^3*b^7+14*a^3*b^6*c-34*a^3*b^5*c^2-44*a^3*b^4*c^3+44*a^3*b^3*c^4+34*a^3*b^2*c^5-14*a^3*b*c^6-4*a^3*c^7-3*a^2*b^8+2*a^2*b^7*c-4*a^2*b^6*c^2+34*a^2*b^5*c^3+86*a^2*b^4*c^4+34*a^2*b^3*c^5-4*a^2*b^2*c^6+2*a^2*b*c^7-3*a^2*c^8-a*b^9-5*a*b^8*c-2*a*b^7*c^2+14*a*b^6*c^3+12*a*b^5*c^4-12*a*b^4*c^5-14*a*b^3*c^6+2*a*b^2*c^7+5*a*b*c^8+a*c^9+b^10+b^9*c-3*b^8*c^2-4*b^7*c^3+2*b^6*c^4+6*b^5*c^5+2*b^4*c^6-4*b^3*c^7-3*b^2*c^8+b*c^9+c^10)=0

数学星空

对以上\(1\sim 4\)结果：

a),做\(R,r,S\)代换得到：

1.   \( 144R^6r^6+72R^5r^7+9R^4r^8-72R^5S^2r^3-990R^4S^2r^4-1536R^3S^2r^5-1152R^2S^2r^6-384RS^2r^7-48S^2r^8+9R^4S^4+384R^2S^4r^2+384RS^4r^3+160S^4r^4-48S^6+(144R^2r^6+72Rr^7+9r^8-72RS^2r^3-126S^2r^4+9S^4)OF^4+(-288R^4r^6-144R^3r^7-18R^2r^8+144R^3S^2r^3+684R^2S^2r^4+384RS^2r^5+96S^2r^6-18R^2S^4-96S^4r^2)OF^2＝0\)

2.   \(81GF^4r^4+16R^2r^6+8Rr^7+r^8-8RS^2r^3-14S^2r^4+18(4Rr^3+r^4-S^2)GF^2r^2+S^4＝0\)

3.   \(9(16R^2r^6+8Rr^7+r^8-8RS^2r^3-14S^2r^4+S^4)HF^4r^4-6(4Rr^3+r^4-S^2)(48R^3r^5+60R^2r^6+24Rr^7+3r^8-12R^2S^2r^2-24RS^2r^3-10S^2r^4+3S^4)HF^2r^2+(48R^3r^5+60R^2r^6+24Rr^7+3r^8-12R^2S^2r^2-24RS^2r^3-10S^2r^4+3S^4)^2＝0\)

4.  \( (144R^2r^6+72Rr^7+9r^8-72RS^2r^3-126S^2r^4+9S^4)IF^4+24(48R^3r^5+24R^2r^6+3Rr^7-12R^2S^2r^2-17RS^2r^3+10S^2r^4+2S^4)IF^2r^2+16(144R^4r^4+72R^3r^5+9R^2r^6-51R^2S^2r^2-12S^2r^4+4S^4)r^4＝0\)

b)，做\(R,r,p\) 代换得到：

1.  \(144R^6r^2-72R^5p^2r+72R^5r^3+9R^4p^4-990R^4p^2r^2+9R^4r^4-1536R^3p^2r^3+384R^2p^4r^2-1152R^2p^2r^4+384Rp^4r^3-384Rp^2r^5-48p^6r^2+160p^4r^4-48p^2r^6+(144R^2r^2-72Rp^2r+72Rr^3+9p^4-126p^2r^2+9r^4)OF^4+(-288R^4r^2+144R^3p^2r-144R^3r^3-18R^2p^4+684R^2p^2r^2-18R^2r^4+384Rp^2r^3-96p^4r^2+96p^2r^4)OF^2=0\)

2.   \(81GF^4+16R^2r^2-8Rp^2r+8Rr^3+p^4-14p^2r^2+r^4+(72Rr-18p^2+18r^2)GF^2=0 \)

3.  \( (144R^2r^2-72Rp^2r+72Rr^3+9p^4-126p^2r^2+9r^4)HF^4-6(4Rr-p^2+r^2)(48R^3r-12R^2p^2+60R^2r^2-24Rp^2r+24Rr^3+3p^4-10p^2r^2+3r^4)HF^2+(48R^3r-12R^2p^2+60R^2r^2-24Rp^2r+24Rr^3+3p^4-10p^2r^2+3r^4)^2=0\)

4.   \((144R^2r^2-72Rp^2r+72Rr^3+9p^4-126p^2r^2+9r^4)IF^4+24(48R^3r-12R^2p^2+24R^2r^2-17Rp^2r+3Rr^3+2p^4+10p^2r^2)IF^2r^2+16(144R^4+72R^3r-51R^2p^2+9R^2r^2+4p^4-12p^2r^2)r^4=0\)

数学星空

关于九点圆圆心\(L\)的心距公式如下：

有关九点圆圆心性质的详细描述见：http://zh.wikipedia.org/wiki/%E4%B9%9D%E7%82%B9%E5%9C%86

1.   \(a^6-a^4b^2-a^4c^2-a^2b^4+3a^2b^2c^2-a^2c^4+b^6-b^4c^2-b^2c^4+c^6+4(a+b-c)(a+b+c)(a-b+c)(a-b-c)OL^2=0\)

2.   \(a^6-a^4b^2-a^4c^2-a^2b^4+3a^2b^2c^2-a^2c^4+b^6-b^4c^2-b^2c^4+c^6+36(a+b-c)(a+b+c)(a-b+c)(a-b-c)GL^2=0\)

3.   \(a^6-a^4b^2-a^4c^2-a^2b^4+3a^2b^2c^2-a^2c^4+b^6-b^4c^2-b^2c^4+c^6+4(a+b-c)(a+b+c)(a-b+c)(a-b-c)HL^2=0\)

4.   \(4(a+b-c)(a+b+c)(a-b+c)(a-b-c)IL^2+(a^3-a^2b-a^2c-ab^2+3abc-ac^2+b^3-b^2c-bc^2+c^3)^2=0\)

5.   \(4(a+b-c)(a+b+c)(a-b+c)(a-b-c)O_1L^2+(a^3+a^2b+a^2c-ab^2+3abc-ac^2-b^3+b^2c+bc^2-c^3)^2=0\)

6.   \(4(a+b-c)(a+b+c)(a-b+c)(a-b-c)O_2L^2+(a^3+a^2b-a^2c-ab^2-3abc-ac^2-b^3-b^2c+bc^2+c^3)^2=0\)

7.   \(4(a+b-c)(a+b+c)(a-b+c)(a-b-c)O_3L^2+(a^3-a^2b+a^2c-ab^2-3abc-ac^2+b^3+b^2c-bc^2-c^3)^2=0\)

8.   \(-24(a+b+c)(a-b-c)(a+b-c)(a-b+c)(a^{10}-3a^8b^2-3a^8c^2+2a^6b^4+2a^6b^2c^2+2a^6c^4+2a^4b^6-a^4b^4c^2-a^4b^2c^4+2a^4c^6-3a^2b^8+2a^2b^6c^2-a^2b^4c^4+2a^2b^2c^6-3a^2c^8+b^{10}-3b^8c^2+2b^6c^4+2b^4c^6-3b^2c^8+c^{10})FL^2+a^{16}+b^{16}+c^{16}+144(a^4-a^2b^2-a^2c^2+b^4-b^2c^2+c^4)(a+b-c)^2(a+b+c)^2(a-b+c)^2(a-b-c)^2FL^4-5a^{14}b^2-5a^{14}c^2+10a^{12}b^4+10a^{12}c^4-11a^{10}b^6-11a^{10}c^6+10a^8b^8+10a^8c^8-11a^6b^{10}-11a^6c^{10}+10a^4b^{12}+10a^4c^{12}-5a^2b^{14}-5a^2c^{14}-5b^{14}c^2+10b^{12}c^4-11b^{10}c^6+10b^8c^8-11b^6c^{10}+10b^4c^{12}-5b^2c^{14}+33a^4b^8c^4+13a^8b^6c^2+13a^6b^8c^2-23a^4b^6c^6+33a^4b^4c^8-27a^4b^2c^{10}+19a^2b^{12}c^2-27a^2b^{10}c^4+13a^2b^8c^6+13a^2b^6c^8-27a^2b^4c^{10}+19a^2b^2c^{12}+13a^6b^2c^8+13a^8b^2c^6-27a^{10}b^4c^2+19a^{12}b^2c^2-23a^6b^4c^6-23a^6b^6c^4-27a^{10}b^2c^4-27a^4b^{10}c^2+33a^8b^4c^4=0\)

数学星空

若做\(R,r,S\)代换，上面\(1\sim 4 \)及\(5\) 式的结果可简化为：

1.   \(4OL^2r^2-9R^2r^2-8Rr^3-2r^4+2S^2=0\)

2.   \(36GL^2r^2-9R^2r^2-8Rr^3-2r^4+2S^2=0\)

3.   \(4HL^2r^2-9R^2r^2-8Rr^3-2r^4+2S^2=0\)

4.   \(-R+2r+2IL=0\)

8.   \((2304R^2r^6+1152Rr^7+144r^8-1152RS^2r^3-2016S^2r^4+144S^4)FL^4+(-1152R^4r^6-576R^3r^7-72R^2r^8+576R^3S^2r^3-2448R^2S^2r^4-3072RS^2r^5-768S^2r^6-72R^2S^4+768S^4r^2)FL^2+144R^6r^6+72R^5r^7+9R^4r^8-72R^5S^2r^3-990R^4S^2r^4-1536R^3S^2r^5-1152R^2S^2r^6-384RS^2r^7-48S^2r^8+9R^4S^4+384R^2S^4r^2+384S^4Rr^3+160S^4r^4-48S^6=0\)


若做\(R,r,p\)代换，上面\(1\sim 4 \)及\(5\) 式的结果可简化为：


1.   \(4OL^2-9R^2-8Rr+2p^2-2r^2=0\)

2.   \(36GL^2-9R^2-8Rr+2p^2-2r^2=0\)

3.   \(4HL^2-9R^2-8Rr+2p^2-2r^2=0\)

4.   \(-R+2r+2IL=0\)

8.   \((2304R^2r^2-1152Rp^2r+1152Rr^3+144p^4-2016p^2r^2+144r^4)FL^4+(-1152R^4r^2+576R^3p^2r-576R^3r^3-72R^2p^4-2448R^2p^2r^2-72R^2r^4-3072Rp^2r^3+768p^4r^2-768p^2r^4)FL^2+144R^6r^2-72R^5p^2r+72R^5r^3+9R^4p^4-990R^4p^2r^2+9R^4r^4-1536R^3p^2r^3+384R^2p^4r^2-1152R^2p^2r^4+384Rp^4r^3-384Rp^2r^5-48p^6r^2+160p^4r^4-48p^2r^6=0\)

数学星空

下一步，我们来讨论http://bbs.emath.ac.cn/thread-2558-1-2.html提到几个问题中的心距公式
第1个分三个周长相等，记为界心\(J\)
第2个分三个内切圆半径\(r_0\)相等，记为等内切圆中心\( E\)
第3个分三个外接圆半径\(R_0\)相等，为垂心\(H\),已讨论