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楼主: 数学星空

[讨论] 三角形两内点间距

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 楼主| 发表于 2014-3-22 10:37:27 | 显示全部楼层
旁心\(O_1,O_2,O_3\)的基本公式如下:
  
\(AO_1 =\sqrt{\frac{pbc}{p-a}},BO_1 =\sqrt{\frac{ac(p-c)}{p-a}},CO_1=\sqrt{\frac{ab(p-b)}{p-a}}\)

\(AO2 =\sqrt{\frac{(p-c)bc}{p-b}},BO2 =\sqrt{\frac{acp}{p-b}}, CO2 =\sqrt{\frac{ab(p-a)}{p-b}}\)

\(AO3 =\sqrt{\frac{(p-b)bc}{p-c}},BO3 =\sqrt{\frac{ac(p-a)}{p-c}},CO3 =\sqrt{\frac{abp}{p-c}}\)

其中\(p=\frac{a+b+c}{2}\)



费马点\(F\)的基本公式如下:

\(AF =\frac{-2a^2+b^2+c^2+k^2}{3k}, BF =\frac{a^2-2b^2+c^2+k^2}{3k}, CF =\frac{a^2+b^2-2c^2+k^2}{3k}\)

其中:\(2k^2=(a^2+b^2+c^2)-\sqrt{3(2a^2b^2+2b^2c^2+2a^2c^2-a^4-b^4-c^4)}\)



九点圆圆心\(L\)的基本公式如下:

\(AL=\frac{\sqrt{a^6-3a^4b^2-3a^4c^2+3a^2b^4+3a^2b^2c^2+3a^2c^4-b^6+b^4c^2+b^2c^4-c^6}}{2\sqrt{-(-c+a+b)(a+b+c)(-b+c+a)(a-b-c)}}\)

\(BL=\frac{\sqrt{-a^6+3a^4b^2+a^4c^2-3a^2b^4+3a^2b^2c^2+a^2c^4+b^6-3b^4c^2+3b^2c^4-c^6}}{2\sqrt{-(-c+a+b)(a+b+c)(-b+c+a)(a-b-c)}}\)

\(CL=\frac{\sqrt{-a^6+a^4b^2+3a^4c^2+a^2b^4+3a^2b^2c^2-3a^2c^4-b^6+3b^4c^2-3b^2c^4+c^6}}{2\sqrt{-(-c+a+b)(a+b+c)(-b+c+a)(a-b-c)}}\)



界心\(J\)的基本公式如下:

\(AJ =\frac{(-a^2+2ab+2ac-b^2+2bc-c^2+8S)(-b+c+a)(a-b-c)(-c+a+b)}{2(5a^4-4a^3b-4a^3c-2a^2b^2+4a^2bc-2a^2c^2-4ab^3+4ab^2c+4abc^2-4ac^3+5b^4-4b^3c-2b^2c^2-4bc^3+5c^4)}-\frac{-a+b+c}{2}\)

\(BJ =\frac{(-a^2+2ab+2ac-b^2+2bc-c^2+8S)(-b+c+a)(a-b-c)(-c+a+b)}{2(5a^4-4a^3b-4a^3c-2a^2b^2+4a^2bc-2a^2c^2-4ab^3+4ab^2c+4abc^2-4ac^3+5b^4-4b^3c-2b^2c^2-4bc^3+5c^4)}-\frac{a-b+c}{2}\)

\(CJ =\frac{(-a^2+2ab+2ac-b^2+2bc-c^2+8S)(-b+c+a)(a-b-c)(-c+a+b)}{2(5a^4-4a^3b-4a^3c-2a^2b^2+4a^2bc-2a^2c^2-4ab^3+4ab^2c+4abc^2-4ac^3+5b^4-4b^3c-2b^2c^2-4bc^3+5c^4)}-\frac{a+b-c}{2}\)

其中:\(S=\frac{\sqrt{2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4}}{4}\)
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2014-3-30 12:17:13 | 显示全部楼层
关于正负等角点\(E_1,E_2\)的心距公式如下

1.   -5*b^2*a^14-5*a^14*c^2+10*b^4*a^12+10*a^12*c^4-11*b^6*a^10-11*a^10*c^6+10*b^8*a^8+10*a^8*c^8-11*b^10*a^6-11*a^6*c^10+10*b^12*a^4+10*a^4*c^12-5*b^14*a^2-5*a^2*c^14-5*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+b^16+a^16+13*a^6*b^8*c^2+13*a^2*b^6*c^8-27*c^4*b^2*a^10+33*c^8*b^4*a^4-23*c^6*b^6*a^4-27*c^2*b^4*a^10+13*a^8*b^2*c^6-27*c^2*b^10*a^4-23*c^6*b^4*a^6+13*a^2*b^8*c^6+19*a^2*b^2*c^12+13*a^8*b^6*c^2-23*c^4*b^6*a^6+13*a^6*b^2*c^8+33*c^4*b^4*a^8+33*c^4*b^8*a^4+19*a^2*b^12*c^2-27*c^10*b^2*a^4+19*a^12*b^2*c^2-27*c^10*b^4*a^2-27*c^4*b^10*a^2+(9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(-c+a+b)^2*(a+b+c)^2*(-b+c+a)^2*(a-b-c)^2*OE1^4+(3*(-c+a+b))*(a+b+c)*(-b+c+a)*(a-b-c)*(a^10-3*a^8*b^2-3*a^8*c^2+2*a^6*b^4+11*a^6*b^2*c^2+2*a^6*c^4+2*a^4*b^6-10*a^4*b^4*c^2-10*a^4*b^2*c^4+2*a^4*c^6-3*a^2*b^8+11*a^2*b^6*c^2-10*a^2*b^4*c^4+11*a^2*b^2*c^6-3*a^2*c^8+b^10-3*b^8*c^2+2*b^6*c^4+2*b^4*c^6-3*b^2*c^8+c^10)*OE1^2=0

2.    81*GE1^4+(-9*a^2-9*b^2-9*c^2)*GE1^2+a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4=0

3.    (9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(-c+a+b)^2*(a+b+c)^2*(-b+c+a)^2*(a-b-c)^2*HE1^4+(3*(a^2+b^2+c^2))*(-c+a+b)*(a+b+c)*(-b+c+a)*(a-b-c)*(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)*HE1^2+(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=0

4.    -a*b^9-a*c^9-a^9*c-a^9*b-b^9*c-b*c^9+2*a^4*b^6-3*b^8*c^2+2*b^4*c^6-3*b^2*c^8-3*a^2*c^8+2*b^6*c^4-3*a^8*b^2+4*a^7*c^3+2*a^6*b^4-6*a^5*b^5-3*a^2*b^8-6*a^5*c^5-3*a^8*c^2+4*a^7*b^3+4*a^3*c^7+4*b^7*c^3-6*b^5*c^5+4*b^3*c^7+2*a^6*c^4+2*a^4*c^6+4*a^3*b^7+c^10+a^10+b^10-7*a^2*b^5*c^3-2*a*b^7*c^2-14*a*b^6*c^3-7*a^2*b^3*c^5-2*a^7*b^2*c-14*a^3*b^6*c-14*a^6*b^3*c+12*a^4*b*c^5+12*a^4*b^5*c+12*a*b^5*c^4+5*a*b^8*c+17*a^4*b^3*c^3-7*a^5*b^3*c^2+17*a^3*b^3*c^4-14*a^6*b*c^3-2*a^7*b*c^2+12*a*b^4*c^5+5*a^8*b*c+12*a^5*b^4*c+5*a*b*c^8-2*a^2*b^7*c-14*a^3*b*c^6-14*a*b^3*c^6-7*a^3*b^5*c^2+17*a^3*b^4*c^3-7*a^5*b^2*c^3-7*a^3*b^2*c^5-2*a*b^2*c^7+12*a^5*b*c^4-2*a^2*b*c^7+23*a^6*b^2*c^2+23*a^2*b^2*c^6-22*a^2*b^4*c^4-22*a^4*b^4*c^2-22*a^4*b^2*c^4+23*a^2*b^6*c^2+(9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a+b+c)^2*IE1^4+(3*(a+b+c))*(a^7-2*a^6*b-2*a^6*c-4*a^5*b^2+6*a^5*b*c-4*a^5*c^2+5*a^4*b^3-a^4*b^2*c-a^4*b*c^2+5*a^4*c^3+5*a^3*b^4-6*a^3*b^3*c+3*a^3*b^2*c^2-6*a^3*b*c^3+5*a^3*c^4-4*a^2*b^5-a^2*b^4*c+3*a^2*b^3*c^2+3*a^2*b^2*c^3-a^2*b*c^4-4*a^2*c^5-2*a*b^6+6*a*b^5*c-a*b^4*c^2-6*a*b^3*c^3-a*b^2*c^4+6*a*b*c^5-2*a*c^6+b^7-2*b^6*c-4*b^5*c^2+5*b^4*c^3+5*b^3*c^4-4*b^2*c^5-2*b*c^6+c^7)*IE1^2=0

5     a*b^9+a*c^9+a^9*c+a^9*b-b^9*c-b*c^9+2*a^4*b^6-3*b^8*c^2+2*b^4*c^6-3*b^2*c^8-3*a^2*c^8+2*b^6*c^4-3*a^8*b^2-4*a^7*c^3+2*a^6*b^4+6*a^5*b^5-3*a^2*b^8+6*a^5*c^5-3*a^8*c^2-4*a^7*b^3-4*a^3*c^7+4*b^7*c^3-6*b^5*c^5+4*b^3*c^7+2*a^6*c^4+2*a^4*c^6-4*a^3*b^7+c^10+a^10+b^10+12*a^4*b^5*c+2*a^7*b*c^2+14*a*b^6*c^3-5*a*b^8*c-14*a^6*b*c^3+2*a^7*b^2*c-12*a*b^5*c^4+17*a^4*b^3*c^3-17*a^3*b^3*c^4+7*a^5*b^3*c^2+5*a^8*b*c-12*a*b^4*c^5-7*a^2*b^5*c^3-7*a^2*b^3*c^5+12*a^4*b*c^5-14*a^6*b^3*c+2*a*b^7*c^2+14*a^3*b^6*c-12*a^5*b^4*c-5*a*b*c^8-2*a^2*b^7*c+14*a^3*b*c^6+14*a*b^3*c^6+7*a^3*b^5*c^2-17*a^3*b^4*c^3+7*a^5*b^2*c^3+7*a^3*b^2*c^5+2*a*b^2*c^7-12*a^5*b*c^4-2*a^2*b*c^7+23*a^6*b^2*c^2+23*a^2*b^2*c^6-22*a^2*b^4*c^4-22*a^4*b^4*c^2-22*a^4*b^2*c^4+23*a^2*b^6*c^2+(9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a-b-c)^2*O1E1^4+(3*(a-b-c))*(a^7+2*a^6*b+2*a^6*c-4*a^5*b^2+6*a^5*b*c-4*a^5*c^2-5*a^4*b^3+a^4*b^2*c+a^4*b*c^2-5*a^4*c^3+5*a^3*b^4-6*a^3*b^3*c+3*a^3*b^2*c^2-6*a^3*b*c^3+5*a^3*c^4+4*a^2*b^5+a^2*b^4*c-3*a^2*b^3*c^2-3*a^2*b^2*c^3+a^2*b*c^4+4*a^2*c^5-2*a*b^6+6*a*b^5*c-a*b^4*c^2-6*a*b^3*c^3-a*b^2*c^4+6*a*b*c^5-2*a*c^6-b^7+2*b^6*c+4*b^5*c^2-5*b^4*c^3-5*b^3*c^4+4*b^2*c^5+2*b*c^6-c^7)*O1E1^2=0

6.   (3*(a-b-c))*(a^7+2*a^6*b+2*a^6*c-4*a^5*b^2+6*a^5*b*c-4*a^5*c^2-5*a^4*b^3+a^4*b^2*c+a^4*b*c^2-5*a^4*c^3+5*a^3*b^4-6*a^3*b^3*c+3*a^3*b^2*c^2-6*a^3*b*c^3+5*a^3*c^4+4*a^2*b^5+a^2*b^4*c-3*a^2*b^3*c^2-3*a^2*b^2*c^3+a^2*b*c^4+4*a^2*c^5-2*a*b^6+6*a*b^5*c-a*b^4*c^2-6*a*b^3*c^3-a*b^2*c^4+6*a*b*c^5-2*a*c^6-b^7+2*b^6*c+4*b^5*c^2-5*b^4*c^3-5*b^3*c^4+4*b^2*c^5+2*b*c^6-c^7)*O1E1^2+(9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a-b-c)^2*O1E1^4+c^10+a^10+b^10+17*a^4*b^3*c^3+2*a^7*b^2*c-17*a^3*b^3*c^4-12*a*b^4*c^5+2*a^7*b*c^2-7*a^2*b^3*c^5+2*a*b^7*c^2-14*a^6*b^3*c+14*a*b^6*c^3-12*a*b^5*c^4+7*a^5*b^3*c^2+12*a^4*b*c^5-5*a*b^8*c-7*a^2*b^5*c^3+14*a^3*b^6*c-12*a^5*b^4*c-5*a*b*c^8-2*a^2*b^7*c+14*a^3*b*c^6+14*a*b^3*c^6+7*a^3*b^5*c^2-17*a^3*b^4*c^3+7*a^5*b^2*c^3+7*a^3*b^2*c^5+2*a*b^2*c^7-12*a^5*b*c^4-2*a^2*b*c^7+23*a^6*b^2*c^2+23*a^2*b^2*c^6-22*a^2*b^4*c^4-22*a^4*b^4*c^2-22*a^4*b^2*c^4+23*a^2*b^6*c^2+12*a^4*b^5*c+5*a^8*b*c-14*a^6*b*c^3+a*b^9+a*c^9+a^9*c+a^9*b-b^9*c-b*c^9+2*a^4*b^6-3*b^8*c^2+2*b^4*c^6-3*b^2*c^8-3*a^2*c^8+2*b^6*c^4-3*a^8*b^2-4*a^7*c^3+2*a^6*b^4+6*a^5*b^5-3*a^2*b^8+6*a^5*c^5-3*a^8*c^2-4*a^7*b^3-4*a^3*c^7+4*b^7*c^3-6*b^5*c^5+4*b^3*c^7+2*a^6*c^4+2*a^4*c^6-4*a^3*b^7=0

7.    -a*b^9+a*c^9+a^9*c-a^9*b+b^9*c+b*c^9+2*a^4*b^6-3*b^8*c^2+2*b^4*c^6-3*b^2*c^8-3*a^2*c^8+2*b^6*c^4-3*a^8*b^2-4*a^7*c^3+2*a^6*b^4-6*a^5*b^5-3*a^2*b^8+6*a^5*c^5-3*a^8*c^2+4*a^7*b^3-4*a^3*c^7-4*b^7*c^3+6*b^5*c^5-4*b^3*c^7+2*a^6*c^4+2*a^4*c^6+4*a^3*b^7+c^10+a^10+b^10-12*a^4*b^5*c+14*a^3*b^6*c-7*a^5*b^3*c^2+14*a*b^6*c^3+2*a^2*b^7*c-5*a^8*b*c+7*a^2*b^5*c^3+12*a*b^5*c^4+14*a^6*b*c^3+17*a^3*b^3*c^4+2*a^7*b^2*c-12*a^4*b*c^5-2*a^7*b*c^2-5*a*b^8*c+14*a^6*b^3*c-12*a*b^4*c^5+7*a^2*b^3*c^5-2*a*b^7*c^2-17*a^4*b^3*c^3-12*a^5*b^4*c+5*a*b*c^8-14*a^3*b*c^6-14*a*b^3*c^6-7*a^3*b^5*c^2-17*a^3*b^4*c^3+7*a^5*b^2*c^3+7*a^3*b^2*c^5+2*a*b^2*c^7+12*a^5*b*c^4+2*a^2*b*c^7+23*a^6*b^2*c^2+23*a^2*b^2*c^6-22*a^2*b^4*c^4-22*a^4*b^4*c^2-22*a^4*b^2*c^4+23*a^2*b^6*c^2+(3*(-c+a+b))*(a^7-2*a^6*b+2*a^6*c-4*a^5*b^2-6*a^5*b*c-4*a^5*c^2+5*a^4*b^3+a^4*b^2*c-a^4*b*c^2-5*a^4*c^3+5*a^3*b^4+6*a^3*b^3*c+3*a^3*b^2*c^2+6*a^3*b*c^3+5*a^3*c^4-4*a^2*b^5+a^2*b^4*c+3*a^2*b^3*c^2-3*a^2*b^2*c^3-a^2*b*c^4+4*a^2*c^5-2*a*b^6-6*a*b^5*c-a*b^4*c^2+6*a*b^3*c^3-a*b^2*c^4-6*a*b*c^5-2*a*c^6+b^7+2*b^6*c-4*b^5*c^2-5*b^4*c^3+5*b^3*c^4+4*b^2*c^5-2*b*c^6-c^7)*O3E1^2+(9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(-c+a+b)^2*O3E1^4=0

8.    -3*a^10*b^2-3*a^10*c^2+3*a^8*b^4+3*a^8*c^4-2*a^6*b^6-2*a^6*c^6+3*a^4*b^8+3*a^4*c^8-3*a^2*b^10-3*a^2*c^10-2*b^6*c^6+3*b^4*c^8-3*b^2*c^10-3*b^10*c^2+3*b^8*c^4+a^12+b^12+c^12-12*a^2*b^6*c^4-12*a^6*b^2*c^4-12*a^6*b^4*c^2-12*a^4*b^6*c^2+15*b^2*a^8*c^2+30*a^4*b^4*c^4+15*a^2*b^8*c^2-12*a^2*b^4*c^6-12*a^4*b^2*c^6+15*a^2*b^2*c^8-(6*(a^2+b^2+c^2))*(a^6*b^2+a^6*c^2-2*a^4*b^4-2*a^4*c^4+a^2*b^6+a^2*c^6+b^6*c^2-2*b^4*c^4+b^2*c^6)*FE1^2+(3*(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*FE1^4=0

9.    -5*a^2*c^14-5*b^14*c^2+10*b^12*c^4-11*b^6*c^10+10*b^8*c^8+10*a^4*b^12-11*a^6*c^10+10*a^8*b^8-11*a^10*c^6-5*a^14*b^2-5*b^2*c^14-11*b^10*c^6-11*a^6*b^10+10*b^4*c^12+10*a^8*c^8+10*a^12*c^4-11*a^10*b^6-5*a^14*c^2+10*a^12*b^4-5*a^2*b^14+10*a^4*c^12+c^16+b^16+a^16+13*a^8*b^6*c^2+19*a^2*b^12*c^2-27*a^10*b^2*c^4-27*a^10*b^4*c^2+13*a^2*b^8*c^6-23*a^6*b^4*c^6-27*a^2*b^4*c^10+19*a^2*b^2*c^12+33*a^4*b^4*c^8-27*a^4*b^10*c^2-23*a^4*b^6*c^6+13*a^6*b^8*c^2+13*a^6*b^2*c^8+33*a^4*b^8*c^4+13*a^2*b^6*c^8+13*a^8*b^2*c^6-27*a^2*b^10*c^4-27*a^4*b^2*c^10-23*a^6*b^6*c^4+19*a^12*b^2*c^2+33*a^8*b^4*c^4+(144*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(-c+a+b)^2*(a+b+c)^2*(-b+c+a)^2*(a-b-c)^2*LE1^4-(24*(-c+a+b))*(a+b+c)*(-b+c+a)*(a-b-c)*(a^10-3*a^8*b^2-3*a^8*c^2+2*a^6*b^4+2*a^6*b^2*c^2+2*a^6*c^4+2*a^4*b^6-a^4*b^4*c^2-a^4*b^2*c^4+2*a^4*c^6-3*a^2*b^8+2*a^2*b^6*c^2-a^2*b^4*c^4+2*a^2*b^2*c^6-3*a^2*c^8+b^10-3*b^8*c^2+2*b^6*c^4+2*b^4*c^6-3*b^2*c^8+c^10)*LE1^2=0

10.  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4-18835552*a^4*b^13*c^15+61179370*a^4*b^12*c^16+2181176*a^4*b^11*c^17-28890412*a^4*b^10*c^18-18268836*a^4*b^9*c^19+28365252*a^4*b^8*c^20+2344112*a^4*b^7*c^21-8964880*a^4*b^6*c^22-2755088*a^4*b^5*c^23+4745468*a^4*b^4*c^24-780996*a^4*b^3*c^25-508412*a^4*b^2*c^26+172784*a^4*b*c^27-8554*a^4*c^28+7896*a^3*b^29-37724*a^3*b^28*c-110888*a^3*b^27*c^2+842080*a^3*b^26*c^3-780996*a^3*b^25*c^4-2911420*a^3*b^24*c^5+4981760*a^3*b^23*c^6+4329024*a^3*b^22*c^7-9597068*a^3*b^21*c^8-10785500*a^3*b^20*c^9+20906960*a^3*b^19*c^10+12073144*a^3*b^18*c^11-22913636*a^3*b^17*c^12-20358192*a^3*b^16*c^13+24354560*a^3*b^15*c^14+24354560*a^3*b^14*c^15-20358192*a^3*b^13*c^16-22913636*a^3*b^12*c^17+12073144*a^3*b^11*c^18+20906960*a^3*b^10*c^19-10785500*a^3*b^9*c^20-9597068*a^3*b^8*c^21+4329024*a^3*b^7*c^22+4981760*a^3*b^6*c^23-2911420*a^3*b^5*c^24-780996*a^3*b^4*c^25+842080*a^3*b^3*c^26-110888*a^3*b^2*c^27-37724*a^3*b*c^28+7896*a^3*c^29-36*a^2*b^30-14604*a^2*b^29*c+94000*a^2*b^28*c^2-110888*a^2*b^27*c^3-508412*a^2*b^26*c^4+1446928*a^2*b^25*c^5-173756*a^2*b^24*c^6-2620784*a^2*b^23*c^7-494028*a^2*b^22*c^8+7811208*a^2*b^21*c^9-3645684*a^2*b^20*c^10-7046392*a^2*b^19*c^11-771136*a^2*b^18*c^12+14910836*a^2*b^17*c^13-4233044*a^2*b^16*c^14-9288416*a^2*b^15*c^15-4233044*a^2*b^14*c^16+14910836*a^2*b^13*c^17-771136*a^2*b^12*c^18-7046392*a^2*b^11*c^19-3645684*a^2*b^10*c^20+7811208*a^2*b^9*c^21-494028*a^2*b^8*c^22-2620784*a^2*b^7*c^23-173756*a^2*b^6*c^24+1446928*a^2*b^5*c^25-508412*a^2*b^4*c^26-110888*a^2*b^3*c^27+94000*a^2*b^2*c^28-14604*a^2*b*c^29-36*a^2*c^30-884*a*b^31+7400*a*b^30*c-14604*a*b^29*c^2-37724*a*b^28*c^3+172784*a*b^27*c^4-68196*a*b^26*c^5-559396*a*b^25*c^6+747568*a*b^24*c^7+478632*a*b^23*c^8-965588*a*b^22*c^9-1253692*a*b^21*c^10+2120856*a*b^20*c^11+1099156*a*b^19*c^12-1985908*a*b^18*c^13-1956816*a*b^17*c^14+2216412*a*b^16*c^15+2216412*a*b^15*c^16-1956816*a*b^14*c^17-1985908*a*b^13*c^18+1099156*a*b^12*c^19+2120856*a*b^11*c^20-1253692*a*b^10*c^21-965588*a*b^9*c^22+478632*a*b^8*c^23+747568*a*b^7*c^24-559396*a*b^6*c^25-68196*a*b^5*c^26+172784*a*b^4*c^27-37724*a*b^3*c^28-14604*a*b^2*c^29+7400*a*b*c^30-884*a*c^31+169*b^32-884*b^31*c-36*b^30*c^2+7896*b^29*c^3-8554*b^28*c^4-42508*b^27*c^5+124472*b^26*c^6-102892*b^25*c^7+6345*b^24*c^8-127876*b^23*c^9+406440*b^22*c^10-333932*b^21*c^11+63398*b^20*c^12-299712*b^19*c^13+694276*b^18*c^14-427196*b^17*c^15+81188*b^16*c^16-427196*b^15*c^17+694276*b^14*c^18-299712*b^13*c^19+63398*b^12*c^20-333932*b^11*c^21+406440*b^10*c^22-127876*b^9*c^23+6345*b^8*c^24-102892*b^7*c^25+124472*b^6*c^26-42508*b^5*c^27-8554*b^4*c^28+7896*b^3*c^29-36*b^2*c^30-884*b*c^31+169*c^32)*JE1=0

11.    b^12*c^4+a^12*b^4+b^4*c^12+a^4*b^12+a^12*c^4+a^4*c^12-4*b^10*c^6-4*a^6*b^10-4*a^10*c^6+6*b^8*c^8-4*a^10*b^6-4*b^6*c^10-4*a^6*c^10+6*a^8*b^8+6*a^8*c^8-a^12*b^2*c^2-a^8*b^6*c^2-a^8*b^2*c^6-a^6*b^8*c^2-a^6*b^2*c^8-a^2*b^12*c^2-a^2*b^8*c^6-a^2*b^6*c^8-a^2*b^2*c^12-4*a^4*b^8*c^4+2*a^10*b^2*c^4-4*a^4*b^4*c^8+2*a^4*b^2*c^10-4*a^8*b^4*c^4+3*a^6*b^4*c^6+2*a^4*b^10*c^2+3*a^6*b^6*c^4+2*a^2*b^10*c^4+2*a^10*b^4*c^2+3*a^4*b^6*c^6+2*a^2*b^4*c^10+(3*(a^2*b^2+a^2*c^2+b^2*c^2))*(a^8*b^2-2*a^8*c^2-4*a^6*b^4-a^6*b^2*c^2+5*a^6*c^4+5*a^4*b^6+a^4*b^4*c^2+a^4*b^2*c^4-4*a^4*c^6-2*a^2*b^8-a^2*b^6*c^2+a^2*b^4*c^4-a^2*b^2*c^6+a^2*c^8+b^8*c^2-4*b^6*c^4+5*b^4*c^6-2*b^2*c^8)*XE1^2+(9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a^2*b^2+a^2*c^2+b^2*c^2)^2*XE1^4=0

12.    a^12*b^4+b^12*c^4+b^4*c^12+a^4*b^12+a^12*c^4+a^4*c^12-4*b^10*c^6-4*a^6*b^10-4*a^10*c^6+6*b^8*c^8-4*a^10*b^6-4*b^6*c^10-4*a^6*c^10+6*a^8*b^8+6*a^8*c^8-a^12*b^2*c^2-a^8*b^6*c^2-a^8*b^2*c^6-a^6*b^8*c^2-a^6*b^2*c^8-a^2*b^12*c^2-a^2*b^8*c^6-a^2*b^6*c^8-a^2*b^2*c^12+3*a^6*b^4*c^6+2*a^2*b^4*c^10+3*a^4*b^6*c^6+2*a^10*b^4*c^2-4*a^8*b^4*c^4+2*a^10*b^2*c^4+3*a^6*b^6*c^4-4*a^4*b^8*c^4+2*a^2*b^10*c^4+2*a^4*b^10*c^2-4*a^4*b^4*c^8+2*a^4*b^2*c^10-(3*(a^2*b^2+a^2*c^2+b^2*c^2))*(2*a^8*b^2-a^8*c^2-5*a^6*b^4+a^6*b^2*c^2+4*a^6*c^4+4*a^4*b^6-a^4*b^4*c^2-a^4*b^2*c^4-5*a^4*c^6-a^2*b^8+a^2*b^6*c^2-a^2*b^4*c^4+a^2*b^2*c^6+2*a^2*c^8+2*b^8*c^2-5*b^6*c^4+4*b^4*c^6-b^2*c^8)*YE1^2+(9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a^2*b^2+a^2*c^2+b^2*c^2)^2*YE1^4=0

13.    (9*a^4-9*a^2*b^2-9*a^2*c^2+9*b^4-9*b^2*c^2+9*c^4)*TE1^4+(3*(a^2+b^2+c^2))*(-c+a+b)*(-b+c+a)*(a+b+c)*(a-b-c)*TE1^2+(-c+a+b)^2*(a+b+c)^2*(-b+c+a)^2*(a-b-c)^2=0

14.     9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)^2*SE1^4+(3*(a^4-4*a^2*b^2-4*a^2*c^2+b^4-4*b^2*c^2+c^4))*(a^6-a^4*b^2-a^4*c^2-a^2*b^4+3*a^2*b^2*c^2-a^2*c^4+b^6-b^4*c^2-b^2*c^4+c^6)*SE1^2+(a^6-a^4*b^2-a^4*c^2-a^2*b^4+3*a^2*b^2*c^2-a^2*c^4+b^6-b^4*c^2-b^2*c^4+c^6)^2=0
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2014-3-30 12:34:55 | 显示全部楼层
1.   10*b^12*c^4-5*b^14*c^2-5*a^2*c^14+10*a^4*c^12-5*a^2*b^14-11*b^10*c^6-5*a^14*c^2+10*a^12*b^4-5*b^2*c^14+10*b^8*c^8+10*a^8*b^8-11*a^10*c^6+10*a^12*c^4-11*a^10*b^6-11*a^6*c^10-5*a^14*b^2-11*a^6*b^10+10*a^8*c^8+10*a^4*b^12+10*b^4*c^12-11*b^6*c^10+b^16+c^16+a^16+33*a^4*b^8*c^4+19*a^12*b^2*c^2-27*a^10*b^4*c^2+13*a^6*b^2*c^8+13*a^8*b^2*c^6+13*a^2*b^6*c^8+33*a^4*b^4*c^8+33*a^8*b^4*c^4-27*a^10*b^2*c^4-23*a^6*b^4*c^6+13*a^8*b^6*c^2-23*a^6*b^6*c^4+13*a^2*b^8*c^6-27*a^2*b^4*c^10-23*a^4*b^6*c^6-27*a^4*b^10*c^2-27*a^4*b^2*c^10-27*a^2*b^10*c^4+19*a^2*b^12*c^2+13*a^6*b^8*c^2+19*a^2*b^2*c^12+(3*(-c+a+b))*(a+b+c)*(-b+c+a)*(a-b-c)*(a^10-3*a^8*b^2-3*a^8*c^2+2*a^6*b^4+11*a^6*b^2*c^2+2*a^6*c^4+2*a^4*b^6-10*a^4*b^4*c^2-10*a^4*b^2*c^4+2*a^4*c^6-3*a^2*b^8+11*a^2*b^6*c^2-10*a^2*b^4*c^4+11*a^2*b^2*c^6-3*a^2*c^8+b^10-3*b^8*c^2+2*b^6*c^4+2*b^4*c^6-3*b^2*c^8+c^10)*OE2^2+(9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(-c+a+b)^2*(a+b+c)^2*(-b+c+a)^2*(a-b-c)^2*OE2^4=0

2.   81*GE2^4+(-9*a^2-9*b^2-9*c^2)*GE2^2+a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4=0

3.  (9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(-c+a+b)^2*(a+b+c)^2*(-b+c+a)^2*(a-b-c)^2*HE2^4+(3*(a^2+b^2+c^2))*(-c+a+b)*(a+b+c)*(-b+c+a)*(a-b-c)*(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)*HE2^2+(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=0

4.   -a*b^9-a*c^9-b*c^9-a^9*c-a^9*b-b^9*c-3*a^2*c^8+2*a^6*b^4+4*a^7*c^3+4*a^7*b^3+2*a^4*c^6+4*a^3*b^7+4*b^7*c^3-6*b^5*c^5+2*a^6*c^4-3*a^2*b^8-6*a^5*c^5-3*b^8*c^2+2*b^6*c^4+2*b^4*c^6-3*b^2*c^8-3*a^8*c^2+2*a^4*b^6+4*b^3*c^7+4*a^3*c^7-6*a^5*b^5-3*a^8*b^2+c^10+a^10+b^10-2*a^2*b^7*c+12*a*b^5*c^4-7*a^5*b^2*c^3-7*a^5*b^3*c^2-22*a^2*b^4*c^4-7*a^2*b^3*c^5+5*a*b*c^8+23*a^6*b^2*c^2-14*a^6*b*c^3-14*a*b^3*c^6-2*a*b^2*c^7-2*a*b^7*c^2+23*a^2*b^2*c^6-2*a^2*b*c^7-22*a^4*b^4*c^2+17*a^4*b^3*c^3+12*a*b^4*c^5+12*a^4*b^5*c-14*a^3*b*c^6-7*a^3*b^2*c^5+17*a^3*b^3*c^4+17*a^3*b^4*c^3-7*a^3*b^5*c^2+12*a^5*b^4*c-2*a^7*b^2*c-2*a^7*b*c^2+12*a^4*b*c^5+5*a^8*b*c+5*a*b^8*c-14*a^6*b^3*c-22*a^4*b^2*c^4-14*a^3*b^6*c+12*a^5*b*c^4-14*a*b^6*c^3-7*a^2*b^5*c^3+23*a^2*b^6*c^2+(3*(a+b+c))*(a^7-2*a^6*b-2*a^6*c-4*a^5*b^2+6*a^5*b*c-4*a^5*c^2+5*a^4*b^3-a^4*b^2*c-a^4*b*c^2+5*a^4*c^3+5*a^3*b^4-6*a^3*b^3*c+3*a^3*b^2*c^2-6*a^3*b*c^3+5*a^3*c^4-4*a^2*b^5-a^2*b^4*c+3*a^2*b^3*c^2+3*a^2*b^2*c^3-a^2*b*c^4-4*a^2*c^5-2*a*b^6+6*a*b^5*c-a*b^4*c^2-6*a*b^3*c^3-a*b^2*c^4+6*a*b*c^5-2*a*c^6+b^7-2*b^6*c-4*b^5*c^2+5*b^4*c^3+5*b^3*c^4-4*b^2*c^5-2*b*c^6+c^7)*IE2^2+(9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a+b+c)^2*IE2^4=0

5.   -12*a*b^5*c^4-14*a^6*b*c^3-2*a^2*b^7*c+5*a^8*b*c+23*a^6*b^2*c^2+2*a*b^7*c^2-12*a*b^4*c^5+17*a^4*b^3*c^3+2*a^7*b*c^2+14*a*b^3*c^6-12*a^5*b^4*c-5*a*b*c^8-22*a^4*b^4*c^2+7*a^5*b^3*c^2+2*a^7*b^2*c+7*a^3*b^5*c^2+12*a^4*b*c^5+14*a^3*b*c^6+7*a^5*b^2*c^3+7*a^3*b^2*c^5-22*a^2*b^4*c^4+12*a^4*b^5*c+23*a^2*b^2*c^6-7*a^2*b^3*c^5-17*a^3*b^4*c^3+2*a*b^2*c^7-17*a^3*b^3*c^4-2*a^2*b*c^7-5*a*b^8*c-14*a^6*b^3*c-22*a^4*b^2*c^4+14*a^3*b^6*c-12*a^5*b*c^4+14*a*b^6*c^3-7*a^2*b^5*c^3+23*a^2*b^6*c^2+a*b^9+a*c^9-b*c^9+a^9*c+a^9*b-b^9*c-3*a^2*c^8+2*a^6*b^4-4*a^7*c^3-4*a^7*b^3+2*a^4*c^6-4*a^3*b^7+4*b^7*c^3-6*b^5*c^5+2*a^6*c^4-3*a^2*b^8+6*a^5*c^5-3*b^8*c^2+2*b^6*c^4+2*b^4*c^6-3*b^2*c^8-3*a^8*c^2+2*a^4*b^6+4*b^3*c^7-4*a^3*c^7+6*a^5*b^5-3*a^8*b^2+c^10+a^10+b^10+(9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a-b-c)^2*O1E2^4+(3*(a-b-c))*(a^7+2*a^6*b+2*a^6*c-4*a^5*b^2+6*a^5*b*c-4*a^5*c^2-5*a^4*b^3+a^4*b^2*c+a^4*b*c^2-5*a^4*c^3+5*a^3*b^4-6*a^3*b^3*c+3*a^3*b^2*c^2-6*a^3*b*c^3+5*a^3*c^4+4*a^2*b^5+a^2*b^4*c-3*a^2*b^3*c^2-3*a^2*b^2*c^3+a^2*b*c^4+4*a^2*c^5-2*a*b^6+6*a*b^5*c-a*b^4*c^2-6*a*b^3*c^3-a*b^2*c^4+6*a*b*c^5-2*a*c^6-b^7+2*b^6*c+4*b^5*c^2-5*b^4*c^3-5*b^3*c^4+4*b^2*c^5+2*b*c^6-c^7)*O1E2^2=0

6.   -7*a^5*b^2*c^3-5*a*b*c^8-12*a^4*b^5*c-17*a^4*b^3*c^3+14*a*b^3*c^6+12*a^5*b^4*c+7*a^3*b^5*c^2-17*a^3*b^3*c^4+23*a^6*b^2*c^2+2*a^2*b*c^7+12*a*b^4*c^5+7*a^2*b^3*c^5-22*a^4*b^4*c^2+23*a^2*b^2*c^6+7*a^5*b^3*c^2+17*a^3*b^4*c^3-22*a^2*b^4*c^4-7*a^3*b^2*c^5-2*a*b^2*c^7+14*a^3*b*c^6-12*a*b^5*c^4+14*a^6*b*c^3+2*a^2*b^7*c+2*a*b^7*c^2-2*a^7*b^2*c+2*a^7*b*c^2-12*a^4*b*c^5-5*a^8*b*c+5*a*b^8*c+14*a^6*b^3*c-22*a^4*b^2*c^4-14*a^3*b^6*c-12*a^5*b*c^4-14*a*b^6*c^3+7*a^2*b^5*c^3+23*a^2*b^6*c^2+a*b^9-a*c^9+b*c^9-a^9*c+a^9*b+b^9*c-3*a^2*c^8+2*a^6*b^4+4*a^7*c^3-4*a^7*b^3+2*a^4*c^6-4*a^3*b^7-4*b^7*c^3+6*b^5*c^5+2*a^6*c^4-3*a^2*b^8-6*a^5*c^5-3*b^8*c^2+2*b^6*c^4+2*b^4*c^6-3*b^2*c^8-3*a^8*c^2+2*a^4*b^6-4*b^3*c^7+4*a^3*c^7+6*a^5*b^5-3*a^8*b^2+c^10+a^10+b^10+(9*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(-b+c+a)^2*O2E2^4+(3*(-b+c+a))*(a^7+2*a^6*b-2*a^6*c-4*a^5*b^2-6*a^5*b*c-4*a^5*c^2-5*a^4*b^3-a^4*b^2*c+a^4*b*c^2+5*a^4*c^3+5*a^3*b^4+6*a^3*b^3*c+3*a^3*b^2*c^2+6*a^3*b*c^3+5*a^3*c^4+4*a^2*b^5-a^2*b^4*c-3*a^2*b^3*c^2+3*a^2*b^2*c^3+a^2*b*c^4-4*a^2*c^5-2*a*b^6-6*a*b^5*c-a*b^4*c^2+6*a*b^3*c^3-a*b^2*c^4-6*a*b*c^5-2*a*c^6-b^7-2*b^6*c+4*b^5*c^2+5*b^4*c^3-5*b^3*c^4-4*b^2*c^5+2*b*c^6+c^7)*O2E2^2=0

7.   -14*a^3*b*c^6-4*a^2*b^2*c^6+86*a^2*b^4*c^4-12*a*b^4*c^5+34*a^2*b^3*c^5-12*a^4*b^5*c+14*a^6*b*c^3-44*a^4*b^3*c^3+34*a^3*b^2*c^5-34*a^3*b^5*c^2+44*a^3*b^3*c^4-2*a*b^7*c^2+5*a*b*c^8-34*a^5*b^3*c^2-44*a^3*b^4*c^3+2*a^2*b^7*c+12*a*b^5*c^4+2*a^2*b*c^7+86*a^4*b^4*c^2+2*a*b^2*c^7-12*a^5*b^4*c-14*a*b^3*c^6+34*a^5*b^2*c^3-4*a^6*b^2*c^2+2*a^7*b^2*c-2*a^7*b*c^2-12*a^4*b*c^5-5*a^8*b*c-5*a*b^8*c+9*(-c+a+b)^2*(a^2+b^2+c^2)^2*O3E2^4+14*a^6*b^3*c+86*a^4*b^2*c^4+14*a^3*b^6*c+12*a^5*b*c^4+14*a*b^6*c^3+34*a^2*b^5*c^3-4*a^2*b^6*c^2-a*b^9+a*c^9+b*c^9+a^9*c-a^9*b+b^9*c-3*a^2*c^8+2*a^6*b^4-4*a^7*c^3+4*a^7*b^3+2*a^4*c^6+4*a^3*b^7-4*b^7*c^3+6*b^5*c^5+2*a^6*c^4-3*a^2*b^8+6*a^5*c^5-3*b^8*c^2+2*b^6*c^4+2*b^4*c^6-3*b^2*c^8-3*a^8*c^2+2*a^4*b^6-4*b^3*c^7-4*a^3*c^7-6*a^5*b^5-3*a^8*b^2+c^10+a^10+b^10+(3*(-c+a+b))*(a^2+b^2+c^2)*(a^5+a^4*b-a^4*c-2*a^3*b^2-6*a^3*b*c-2*a^3*c^2-2*a^2*b^3+14*a^2*b^2*c-14*a^2*b*c^2+2*a^2*c^3+a*b^4-6*a*b^3*c-14*a*b^2*c^2-6*a*b*c^3+a*c^4+b^5-b^4*c-2*b^3*c^2+2*b^2*c^3+b*c^4-c^5)*O3E2^2=0

8.   -11*b^6*c^10+10*b^4*c^12-11*b^10*c^6+10*b^8*c^8-5*b^14*c^2+10*b^12*c^4+10*a^12*b^4-5*a^14*b^2-5*a^14*c^2-11*a^6*c^10+10*a^4*c^12+10*a^4*b^12-5*a^2*c^14-5*a^2*b^14-11*a^10*c^6+10*a^8*b^8+10*a^8*c^8-11*a^6*b^10-11*a^10*b^6+10*a^12*c^4-5*b^2*c^14+c^16+a^16+b^16+33*a^4*b^4*c^8-27*a^2*b^10*c^4-27*a^10*b^4*c^2+33*a^8*b^4*c^4-23*a^6*b^6*c^4+13*a^2*b^8*c^6-27*a^2*b^4*c^10-27*a^10*b^2*c^4-27*a^4*b^2*c^10-23*a^6*b^4*c^6-27*a^4*b^10*c^2+13*a^6*b^8*c^2+13*a^6*b^2*c^8+13*a^2*b^6*c^8+19*a^12*b^2*c^2+19*a^2*b^12*c^2+13*a^8*b^6*c^2+19*a^2*b^2*c^12+13*a^8*b^2*c^6+33*a^4*b^8*c^4-23*a^4*b^6*c^6-(24*(-c+a+b))*(a+b+c)*(c+a-b)*(-c+a-b)*(a^10-3*a^8*b^2-3*a^8*c^2+2*a^6*b^4+2*a^6*b^2*c^2+2*a^6*c^4+2*a^4*b^6-a^4*b^4*c^2-a^4*b^2*c^4+2*a^4*c^6-3*a^2*b^8+2*a^2*b^6*c^2-a^2*b^4*c^4+2*a^2*b^2*c^6-3*a^2*c^8+b^10-3*b^8*c^2+2*b^6*c^4+2*b^4*c^6-3*b^2*c^8+c^10)*LE2^2+(144*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(-c+a+b)^2*(a+b+c)^2*(c+a-b)^2*(-c+a-b)^2*LE2^4=0

9.  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4-18835552*a^4*b^13*c^15+61179370*a^4*b^12*c^16+2181176*a^4*b^11*c^17-28890412*a^4*b^10*c^18-18268836*a^4*b^9*c^19+28365252*a^4*b^8*c^20+2344112*a^4*b^7*c^21-8964880*a^4*b^6*c^22-2755088*a^4*b^5*c^23+4745468*a^4*b^4*c^24-780996*a^4*b^3*c^25-508412*a^4*b^2*c^26+172784*a^4*b*c^27-8554*a^4*c^28+7896*a^3*b^29-37724*a^3*b^28*c-110888*a^3*b^27*c^2+842080*a^3*b^26*c^3-780996*a^3*b^25*c^4-2911420*a^3*b^24*c^5+4981760*a^3*b^23*c^6+4329024*a^3*b^22*c^7-9597068*a^3*b^21*c^8-10785500*a^3*b^20*c^9+20906960*a^3*b^19*c^10+12073144*a^3*b^18*c^11-22913636*a^3*b^17*c^12-20358192*a^3*b^16*c^13+24354560*a^3*b^15*c^14+24354560*a^3*b^14*c^15-20358192*a^3*b^13*c^16-22913636*a^3*b^12*c^17+12073144*a^3*b^11*c^18+20906960*a^3*b^10*c^19-10785500*a^3*b^9*c^20-9597068*a^3*b^8*c^21+4329024*a^3*b^7*c^22+4981760*a^3*b^6*c^23-2911420*a^3*b^5*c^24-780996*a^3*b^4*c^25+842080*a^3*b^3*c^26-110888*a^3*b^2*c^27-37724*a^3*b*c^28+7896*a^3*c^29-36*a^2*b^30-14604*a^2*b^29*c+94000*a^2*b^28*c^2-110888*a^2*b^27*c^3-508412*a^2*b^26*c^4+1446928*a^2*b^25*c^5-173756*a^2*b^24*c^6-2620784*a^2*b^23*c^7-494028*a^2*b^22*c^8+7811208*a^2*b^21*c^9-3645684*a^2*b^20*c^10-7046392*a^2*b^19*c^11-771136*a^2*b^18*c^12+14910836*a^2*b^17*c^13-4233044*a^2*b^16*c^14-9288416*a^2*b^15*c^15-4233044*a^2*b^14*c^16+14910836*a^2*b^13*c^17-771136*a^2*b^12*c^18-7046392*a^2*b^11*c^19-3645684*a^2*b^10*c^20+7811208*a^2*b^9*c^21-494028*a^2*b^8*c^22-2620784*a^2*b^7*c^23-173756*a^2*b^6*c^24+1446928*a^2*b^5*c^25-508412*a^2*b^4*c^26-110888*a^2*b^3*c^27+94000*a^2*b^2*c^28-14604*a^2*b*c^29-36*a^2*c^30-884*a*b^31+7400*a*b^30*c-14604*a*b^29*c^2-37724*a*b^28*c^3+172784*a*b^27*c^4-68196*a*b^26*c^5-559396*a*b^25*c^6+747568*a*b^24*c^7+478632*a*b^23*c^8-965588*a*b^22*c^9-1253692*a*b^21*c^10+2120856*a*b^20*c^11+1099156*a*b^19*c^12-1985908*a*b^18*c^13-1956816*a*b^17*c^14+2216412*a*b^16*c^15+2216412*a*b^15*c^16-1956816*a*b^14*c^17-1985908*a*b^13*c^18+1099156*a*b^12*c^19+2120856*a*b^11*c^20-1253692*a*b^10*c^21-965588*a*b^9*c^22+478632*a*b^8*c^23+747568*a*b^7*c^24-559396*a*b^6*c^25-68196*a*b^5*c^26+172784*a*b^4*c^27-37724*a*b^3*c^28-14604*a*b^2*c^29+7400*a*b*c^30-884*a*c^31+169*b^32-884*b^31*c-36*b^30*c^2+7896*b^29*c^3-8554*b^28*c^4-42508*b^27*c^5+124472*b^26*c^6-102892*b^25*c^7+6345*b^24*c^8-127876*b^23*c^9+406440*b^22*c^10-333932*b^21*c^11+63398*b^20*c^12-299712*b^19*c^13+694276*b^18*c^14-427196*b^17*c^15+81188*b^16*c^16-427196*b^15*c^17+694276*b^14*c^18-299712*b^13*c^19+63398*b^12*c^20-333932*b^11*c^21+406440*b^10*c^22-127876*b^9*c^23+6345*b^8*c^24-102892*b^7*c^25+124472*b^6*c^26-42508*b^5*c^27-8554*b^4*c^28+7896*b^3*c^29-36*b^2*c^30-884*b*c^31+169*c^32)*JE1=0
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2014-3-30 12:35:18 | 显示全部楼层
关于斯坦纳\(点F_1,F_2\)现得到部分心距公式如下:

注:\(F_1,F_2\)满足同一个方程

1.   c^4*a^4*b^4*(a^6-a^4*b^2-a^4*c^2-a^2*b^4+3*a^2*b^2*c^2-a^2*c^4+b^6-b^4*c^2-b^2*c^4+c^6)^2+4*c^2*a^2*b^2*(-c+a+b)*(a+b+c)*(-b+c+a)*(a-b-c)*(a^6-a^4*b^2-a^4*c^2-a^2*b^4+3*a^2*b^2*c^2-a^2*c^4+b^6-b^4*c^2-b^2*c^4+c^6)^2*OF1^2+(6*(2*a^10*b^2+2*a^10*c^2-8*a^8*b^4+a^8*b^2*c^2-8*a^8*c^4+12*a^6*b^6-3*a^6*b^4*c^2-3*a^6*b^2*c^4+12*a^6*c^6-8*a^4*b^8-3*a^4*b^6*c^2+15*a^4*b^4*c^4-3*a^4*b^2*c^6-8*a^4*c^8+2*a^2*b^10+a^2*b^8*c^2-3*a^2*b^6*c^4-3*a^2*b^4*c^6+a^2*b^2*c^8+2*a^2*c^10+2*b^10*c^2-8*b^8*c^4+12*b^6*c^6-8*b^4*c^8+2*b^2*c^10))*(-c+a+b)^2*(a+b+c)^2*(-b+c+a)^2*(a-b-c)^2*OF1^4+(36*(a^6-a^4*b^2-a^4*c^2-a^2*b^4+3*a^2*b^2*c^2-a^2*c^4+b^6-b^4*c^2-b^2*c^4+c^6))*(-c+a+b)^3*(a+b+c)^3*(-b+c+a)^3*(a-b-c)^3*OF1^6+81*(-c+a+b)^4*(a+b+c)^4*(-b+c+a)^4*(a-b-c)^4*OF1^8=0

2.   81*GF1^4-a^4+a^2*b^2+a^2*c^2-b^4+b^2*c^2-c^4=0

3.   (a^12-a^10*b^2-a^10*c^2-5*a^8*b^4+11*a^8*b^2*c^2-5*a^8*c^4+10*a^6*b^6-10*a^6*b^4*c^2-10*a^6*b^2*c^4+10*a^6*c^6-5*a^4*b^8-10*a^4*b^6*c^2+30*a^4*b^4*c^4-10*a^4*b^2*c^6-5*a^4*c^8-a^2*b^10+11*a^2*b^8*c^2-10*a^2*b^6*c^4-10*a^2*b^4*c^6+11*a^2*b^2*c^8-a^2*c^10+b^12-b^10*c^2-5*b^8*c^4+10*b^6*c^6-5*b^4*c^8-b^2*c^10+c^12)^2+(16*(a^6-a^4*b^2-a^4*c^2-a^2*b^4+3*a^2*b^2*c^2-a^2*c^4+b^6-b^4*c^2-b^2*c^4+c^6))*(a^12-a^10*b^2-a^10*c^2-5*a^8*b^4+11*a^8*b^2*c^2-5*a^8*c^4+10*a^6*b^6-10*a^6*b^4*c^2-10*a^6*b^2*c^4+10*a^6*c^6-5*a^4*b^8-10*a^4*b^6*c^2+30*a^4*b^4*c^4-10*a^4*b^2*c^6-5*a^4*c^8-a^2*b^10+11*a^2*b^8*c^2-10*a^2*b^6*c^4-10*a^2*b^4*c^6+11*a^2*b^2*c^8-a^2*c^10+b^12-b^10*c^2-5*b^8*c^4+10*b^6*c^6-5*b^4*c^8-b^2*c^10+c^12)*(-c+a+b)*(a+b+c)*(-b+c+a)*(a-b-c)*HF1^2+(6*(13*a^12-21*a^10*b^2-21*a^10*c^2-33*a^8*b^4+111*a^8*b^2*c^2-33*a^8*c^4+82*a^6*b^6-90*a^6*b^4*c^2-90*a^6*b^2*c^4+82*a^6*c^6-33*a^4*b^8-90*a^4*b^6*c^2+246*a^4*b^4*c^4-90*a^4*b^2*c^6-33*a^4*c^8-21*a^2*b^10+111*a^2*b^8*c^2-90*a^2*b^6*c^4-90*a^2*b^4*c^6+111*a^2*b^2*c^8-21*a^2*c^10+13*b^12-21*b^10*c^2-33*b^8*c^4+82*b^6*c^6-33*b^4*c^8-21*b^2*c^10+13*c^12))*(-c+a+b)^2*(a+b+c)^2*(-b+c+a)^2*(a-b-c)^2*HF1^4+(144*(a^6-a^4*b^2-a^4*c^2-a^2*b^4+3*a^2*b^2*c^2-a^2*c^4+b^6-b^4*c^2-b^2*c^4+c^6))*(-c+a+b)^3*(a+b+c)^3*(-b+c+a)^3*(a-b-c)^3*HF1^6+81*(-c+a+b)^4*(a+b+c)^4*(-b+c+a)^4*(a-b-c)^4*HF1^8=0

4.   16*a^2*b^2*c^2*(a^3-a^2*b-a^2*c-a*b^2+3*a*b*c-a*c^2+b^3-b^2*c-b*c^2+c^3)^2+16*b*c*a*(a+b+c)*(a^3-a^2*b-a^2*c-a*b^2+3*a*b*c-a*c^2+b^3-b^2*c-b*c^2+c^3)*(a^3-2*a^2*b-2*a^2*c-2*a*b^2+9*a*b*c-2*a*c^2+b^3-2*b^2*c-2*b*c^2+c^3)*IF1^2-(24*(a^5*b+a^5*c-7*a^4*b*c-2*a^3*b^3+8*a^3*b^2*c+8*a^3*b*c^2-2*a^3*c^3+8*a^2*b^3*c-27*a^2*b^2*c^2+8*a^2*b*c^3+a*b^5-7*a*b^4*c+8*a*b^3*c^2+8*a*b^2*c^3-7*a*b*c^4+a*c^5+b^5*c-2*b^3*c^3+b*c^5))*(a+b+c)^2*IF1^4+(36*(a^3-2*a^2*b-2*a^2*c-2*a*b^2+9*a*b*c-2*a*c^2+b^3-2*b^2*c-2*b*c^2+c^3))*(a+b+c)^3*IF1^6+81*(a+b+c)^4*IF1^8=0

5.   16*c^2*b^2*a^2*(a^3+a^2*b+a^2*c-a*b^2+3*a*b*c-a*c^2-b^3+b^2*c+b*c^2-c^3)^2+16*b*c*a*(a-b-c)*(a^3+2*a^2*b+2*a^2*c-2*a*b^2+9*a*b*c-2*a*c^2-b^3+2*b^2*c+2*b*c^2-c^3)*(a^3+a^2*b+a^2*c-a*b^2+3*a*b*c-a*c^2-b^3+b^2*c+b*c^2-c^3)*O1F1^2+(24*(a^5*b+a^5*c+7*a^4*b*c-2*a^3*b^3+8*a^3*b^2*c+8*a^3*b*c^2-2*a^3*c^3-8*a^2*b^3*c+27*a^2*b^2*c^2-8*a^2*b*c^3+a*b^5-7*a*b^4*c+8*a*b^3*c^2+8*a*b^2*c^3-7*a*b*c^4+a*c^5-b^5*c+2*b^3*c^3-b*c^5))*(a-b-c)^2*O1F1^4+(36*(a^3+2*a^2*b+2*a^2*c-2*a*b^2+9*a*b*c-2*a*c^2-b^3+2*b^2*c+2*b*c^2-c^3))*(a-b-c)^3*O1F1^6+81*(a-b-c)^4*O1F1^8=0

6.   16*a^2*b^2*c^2*(a^3+a^2*b-a^2*c-a*b^2-3*a*b*c-a*c^2-b^3-b^2*c+b*c^2+c^3)^2-16*b*c*a*(-b+c+a)*(a^3+a^2*b-a^2*c-a*b^2-3*a*b*c-a*c^2-b^3-b^2*c+b*c^2+c^3)*(a^3+2*a^2*b-2*a^2*c-2*a*b^2-9*a*b*c-2*a*c^2-b^3-2*b^2*c+2*b*c^2+c^3)*O2F1^2+(24*(a^5*b-a^5*c-7*a^4*b*c-2*a^3*b^3-8*a^3*b^2*c+8*a^3*b*c^2+2*a^3*c^3+8*a^2*b^3*c+27*a^2*b^2*c^2+8*a^2*b*c^3+a*b^5+7*a*b^4*c+8*a*b^3*c^2-8*a*b^2*c^3-7*a*b*c^4-a*c^5+b^5*c-2*b^3*c^3+b*c^5))*(-b+c+a)^2*O2F1^4+(36*(a^3+2*a^2*b-2*a^2*c-2*a*b^2-9*a*b*c-2*a*c^2-b^3-2*b^2*c+2*b*c^2+c^3))*(-b+c+a)^3*O2F1^6+81*(-b+c+a)^4*O2F1^8=0

7.    16*c^2*b^2*a^2*(a^3-a^2*b+a^2*c-a*b^2-3*a*b*c-a*c^2+b^3+b^2*c-b*c^2-c^3)^2-16*b*c*a*(-c+a+b)*(a^3-a^2*b+a^2*c-a*b^2-3*a*b*c-a*c^2+b^3+b^2*c-b*c^2-c^3)*(a^3-2*a^2*b+2*a^2*c-2*a*b^2-9*a*b*c-2*a*c^2+b^3+2*b^2*c-2*b*c^2-c^3)*O3F1^2-(24*(a^5*b-a^5*c+7*a^4*b*c-2*a^3*b^3-8*a^3*b^2*c+8*a^3*b*c^2+2*a^3*c^3-8*a^2*b^3*c-27*a^2*b^2*c^2-8*a^2*b*c^3+a*b^5+7*a*b^4*c+8*a*b^3*c^2-8*a*b^2*c^3-7*a*b*c^4-a*c^5-b^5*c+2*b^3*c^3-b*c^5))*(-c+a+b)^2*O3F1^4+(36*(a^3-2*a^2*b+2*a^2*c-2*a*b^2-9*a*b*c-2*a*c^2+b^3+2*b^2*c-2*b*c^2-c^3))*(-c+a+b)^3*O3F1^6+81*(-c+a+b)^4*O3F1^8=0

8.   (a^6-a^4*b^2-a^4*c^2-a^2*b^4+3*a^2*b^2*c^2-a^2*c^4+b^6-b^4*c^2-b^2*c^4+c^6)^4-(24*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a^6-a^4*b^2-a^4*c^2-a^2*b^4+3*a^2*b^2*c^2-a^2*c^4+b^6-b^4*c^2-b^2*c^4+c^6)^3*FF1^2+(18*(17*a^8-34*a^6*b^2-34*a^6*c^2+42*a^4*b^4+9*a^4*b^2*c^2+42*a^4*c^4-34*a^2*b^6+9*a^2*b^4*c^2+9*a^2*b^2*c^4-34*a^2*c^6+17*b^8-34*b^6*c^2+42*b^4*c^4-34*b^2*c^6+17*c^8))*(a^6-a^4*b^2-a^4*c^2-a^2*b^4+3*a^2*b^2*c^2-a^2*c^4+b^6-b^4*c^2-b^2*c^4+c^6)^2*FF1^4-(324*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4))*(a^6-a^4*b^2-a^4*c^2-a^2*b^4+3*a^2*b^2*c^2-a^2*c^4+b^6-b^4*c^2-b^2*c^4+c^6)*(7*a^8-12*a^6*b^2-12*a^6*c^2+10*a^4*b^4+7*a^4*b^2*c^2+10*a^4*c^4-12*a^2*b^6+7*a^2*b^4*c^2+7*a^2*b^2*c^4-12*a^2*c^6+7*b^8-12*b^6*c^2+10*b^4*c^4-12*b^2*c^6+7*c^8)*FF1^6+(9963*a^16-31590*a^14*b^2-31590*a^14*c^2+54432*a^12*b^4+71928*a^12*b^2*c^2+54432*a^12*c^4-73386*a^10*b^6-75816*a^10*b^4*c^2-75816*a^10*b^2*c^4-73386*a^10*c^6+81162*a^8*b^8+31590*a^8*b^6*c^2+75087*a^8*b^4*c^4+31590*a^8*b^2*c^6+81162*a^8*c^8-73386*a^6*b^10+31590*a^6*b^8*c^2-48600*a^6*b^6*c^4-48600*a^6*b^4*c^6+31590*a^6*b^2*c^8-73386*a^6*c^10+54432*a^4*b^12-75816*a^4*b^10*c^2+75087*a^4*b^8*c^4-48600*a^4*b^6*c^6+75087*a^4*b^4*c^8-75816*a^4*b^2*c^10+54432*a^4*c^12-31590*a^2*b^14+71928*a^2*b^12*c^2-75816*a^2*b^10*c^4+31590*a^2*b^8*c^6+31590*a^2*b^6*c^8-75816*a^2*b^4*c^10+71928*a^2*b^2*c^12-31590*a^2*c^14+9963*b^16-31590*b^14*c^2+54432*b^12*c^4-73386*b^10*c^6+81162*b^8*c^8-73386*b^6*c^10+54432*b^4*c^12-31590*b^2*c^14+9963*c^16)*FF1^8-(26244*(a^4+b^4+c^4))*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(a^6-a^4*b^2-a^4*c^2-a^2*b^4+3*a^2*b^2*c^2-a^2*c^4+b^6-b^4*c^2-b^2*c^4+c^6)*FF1^10+(13122*(a^2+b^2+c^2))*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)*(5*a^6-3*a^4*b^2-3*a^4*c^2-3*a^2*b^4+3*a^2*b^2*c^2-3*a^2*c^4+5*b^6-3*b^4*c^2-3*b^2*c^4+5*c^6)*FF1^12-(236196*(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*FF1^14+531441*(a^4-a^2*b^2-a^2*c^2+b^4-b^2*c^2+c^4)^2*FF1^16=0
   
           
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2014-4-13 21:30:57 | 显示全部楼层
对于30#提到了的:

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

第2个问题,我们得到了\(r\)的代数方程:\(A_1\)

\((a+b+c)^4(-c+a+b)^6(-c+a-b)^6(c+a-b)^6+8(9a^2+10ab+10ac+9b^2+10bc+9c^2)(a+b+c)^3(-c+a+b)^5(-c+a-b)^5(c+a-b)^5r^2+16(125a^4+268a^3b+268a^3c+382a^2b^2+580a^2bc+382a^2c^2+268ab^3+580ab^2c+580abc^2+268ac^3+125b^4+268b^3c+382b^2c^2+
268bc^3+125c^4)(a+b+c)^2(-c+a+b)^4(-c+a-b)^4(c+a-b)^4r^4+1024(a+b+c)(24a^6+73a^5b+73a^5c+152a^4b^2+248a^4bc+152a^4c^2+
190a^3b^3+447a^3b^2c+447a^3bc^2+190a^3c^3+152a^2b^4+447a^2b^3c+630a^2b^2c^2+447a^2bc^3+152a^2c^4+73ab^5+
248ab^4c+447ab^3c^2+447ab^2c^3+248abc^4+73ac^5+24b^6+73b^5c+152b^4c^2+190b^3c^3+152b^2c^4+73bc^5+24c^6)(-c+a+b)^3
(-c+a-b)^3(c+a-b)^3r^6+1024(101a^8+374a^7b+374a^7c+1244a^6b^2+2018a^6bc+1244a^6c^2+2426a^5b^3+5814a^5b^2c+
5814a^5bc^2+2426a^5c^3+2974a^4b^4+9202a^4b^3c+13188a^4b^2c^2+9202a^4bc^3+2974a^4c^4+2426a^3b^5+9202a^3b^4c+
17012a^3b^3c^2+17012a^3b^2c^3+9202a^3bc^4+2426a^3c^5+1244a^2b^6+5814a^2b^5c+13188a^2b^4c^2+17012a^2b^3c^3+
13188a^2b^2c^4+5814a^2bc^5+1244a^2c^6+374ab^7+2018ab^6c+5814ab^5c^2+9202ab^4c^3+9202ab^3c^4+5814ab^2c^5+
2018abc^6+374ac^7+101b^8+374b^7c+1244b^6c^2+2426b^5c^3+2974b^4c^4+2426b^3c^5+1244b^2c^6+374bc^7+101c^8)(-c+a+b)^2
(-c+a-b)^2(c+a-b)^2r^8-16384(-c+a+b)(c+a-b)(-c+a-b)(6a^9+33a^8b+33a^8c-129a^7b^2+24a^7bc-129a^7c^2-383a^6b^3-474a^6b^2c-
474a^6bc^2-383a^6c^3-551a^5b^4-1808a^5b^3c-1944a^5b^2c^2-1808a^5bc^3-551a^5c^4-551a^4b^5-2718a^4b^4c-4109a^4b^3c^2-
4109a^4b^2c^3-2718a^4bc^4-551a^4c^5-383a^3b^6-1808a^3b^5c-4109a^3b^4c^2-4920a^3b^3c^3-4109a^3b^2c^4-1808a^3bc^5-
383a^3c^6-129a^2b^7-474a^2b^6c-1944a^2b^5c^2-4109a^2b^4c^3-4109a^2b^3c^4-1944a^2b^2c^5-474a^2bc^6-129a^2c^7+33ab^8+
24ab^7c-474ab^6c^2-1808ab^5c^3-2718ab^4c^4-1808ab^3c^5-474ab^2c^6+24abc^7+33ac^8+6b^9+33b^8c-129b^7c^2-383b^6c^3-
551b^5c^4-551b^4c^5-383b^3c^6-129b^2c^7+33bc^8+6c^9)r^{10}+(262144a^{10}+262144a^9b+262144a^9c-3866624a^8b^2-262144a^8bc-
3866624a^8c^2-20054016a^7b^3-9306112a^7b^2c-9306112a^7bc^2-20054016a^7c^3+3604480a^6b^4-2490368a^6b^3c+65732608a^6b^2c^2-
2490368a^6bc^3+3604480a^6c^4+39583744a^5b^5+11796480a^5b^4c+91881472a^5b^3c^2+91881472a^5b^2c^3+11796480a^5bc^4+
39583744a^5c^5+3604480a^4b^6+11796480a^4b^5c+80216064a^4b^4c^2+362676224a^4b^3c^3+80216064a^4b^2c^4+11796480a^4bc^5+
3604480a^4c^6-20054016a^3b^7-2490368a^3b^6c+91881472a^3b^5c^2+362676224a^3b^4c^3+362676224a^3b^3c^4+91881472a^3b^2c^5-
2490368a^3bc^6-20054016a^3c^7-3866624a^2b^8-9306112a^2b^7c+65732608a^2b^6c^2+91881472a^2b^5c^3+80216064a^2b^4c^4+
91881472a^2b^3c^5+65732608a^2b^2c^6-9306112a^2bc^7-3866624a^2c^8+262144ab^9-262144ab^8c-9306112ab^7c^2-2490368ab^6c^3+
11796480ab^5c^4+11796480ab^4c^5-2490368ab^3c^6-9306112ab^2c^7-262144abc^8+262144ac^9+262144b^{10}+262144b^9c-3866624b^8c^2-20054016b^7c^3+3604480b^6c^4+39583744b^5c^5+3604480b^4c^6-20054016b^3c^7-3866624b^2c^8+262144bc^9+
262144c^10)r^{12}+(9437184a^6b^2+9437184a^6c^2+9437184a^5b^2c+9437184a^5bc^2-18874368a^4b^4-9437184a^4b^3c-
113246208a^4b^2c^2-9437184a^4bc^3-18874368a^4c^4-9437184a^3b^4c-452984832a^3b^3c^2-452984832a^3b^2c^3-9437184a^3bc^4+
9437184a^2b^6+9437184a^2b^5c-113246208a^2b^4c^2-452984832a^2b^3c^3-113246208a^2b^2c^4+9437184a^2bc^5+9437184a^2c^6+
9437184ab^5c^2-9437184ab^4c^3-9437184ab^3c^4+9437184ab^2c^5+9437184b^6c^2-18874368b^4c^4+9437184b^2c^6)r^{14}+339738624a^2b^2c^2r^{16}=0 \)


并得到了\(AE+BE+CE=x+y+z=L\)的代数方程:\(A_2\)

\((c+a+b)^2(a^3-a^2b-a^2c-ab^2+6abc-ac^2+b^3-b^2c-bc^2+c^3)^2-(c+a+b)(11a^5-5a^4b-5a^4c+10a^3b^2+8a^3bc+10a^3c^2+10a^2b^3+10a^2b^2c+10a^2bc^2+10a^2c^3-5ab^4+8ab^3c+10ab^2c^2+8abc^3-5ac^4+11b^5-5b^4c+10b^3c^2+10b^2c^3-5bc^4+11c^5)L^2+(44a^4+32a^3b+32a^3c+40a^2b^2+32a^2bc+40a^2c^2+32ab^3+32ab^2c+32abc^2+32ac^3+44b^4+32b^3c+40b^2c^2+32bc^3+44c^4)L^4+(-80a^2-32ab-32ac-80b^2-32bc-80c^2)L^6+64L^8=0\)



通过千辛万苦的计算得到:

\(bc(-c+a-b)(a^3+3a^2b+3a^2c-ab^2+6abc-ac^2-3b^3-9b^2c-9bc^2-3c^3)(a^4-2a^3b-2a^3c-7a^2b^2-6a^2bc-7a^2c^2+2ab^3+22ab^2c+22abc^2+2ac^3+6b^4+36b^3c+60b^2c^2+36bc^3+6c^4)+(2(a+b+c))(-c+a-b)(2a^6b+2a^6c-4a^5b^2+26a^5bc-4a^5c^2-16a^4b^3-31a^4b^2c-31a^4bc^2-16a^4c^3+8a^3b^4-122a^3b^3c-204a^3b^2c^2-122a^3bc^3+8a^3c^4+26a^2b^5+58a^2b^4c+24a^2b^3c^2+24a^2b^2c^3+58a^2bc^4+26a^2c^5-4ab^6+112ab^5c+332ab^4c^2+272ab^3c^3+332ab^2c^4+112abc^5-4ac^6-12b^7-45b^6c-117b^5c^2-258b^4c^3-258b^3c^4-117b^2c^5-45bc^6-12c^7)x+(16a^8+80a^7b+80a^7c-40a^6b^2+241a^6bc-40a^6c^2-368a^5b^3-398a^5b^2c-398a^5bc^2-368a^5c^3-40a^4b^4-697a^4b^3c-710a^4b^2c^2-697a^4bc^3-40a^4c^4+496a^3b^5+908a^3b^4c+1748a^3b^3c^2+1748a^3b^2c^3+908a^3bc^4+496a^3c^5+136a^2b^6+895a^2b^5c+2796a^2b^4c^2+4538a^2b^3c^3+2796a^2b^2c^4+895a^2bc^5+136a^2c^6-208ab^7-462ab^6c-966ab^5c^2-2108ab^4c^3-2108ab^3c^4-966ab^2c^5-462abc^6-208ac^7-72b^8-567b^7c-2430b^6c^2-5913b^5c^3-7956b^4c^4-5913b^3c^5-2430b^2c^6-567bc^7-72c^8)x^2+(4(a+b+c))(16a^6+45a^5b+45a^5c-29a^4b^2-168a^4bc-29a^4c^2-202a^3b^3-494a^3b^2c-494a^3bc^2-202a^3c^3-6a^2b^4+332a^2b^3c-292a^2b^2c^2+332a^2bc^3-6a^2c^4+221ab^5+561ab^4c+1234ab^3c^2+1234ab^2c^3+561abc^4+221ac^5-45b^6-276b^5c-579b^4c^2-792b^3c^3-579b^2c^4-276bc^5-45c^6)x^3+(656a^6+1344a^5b+1344a^5c-368a^4b^2+1900a^4bc-368a^4c^2-1664a^3b^3-2528a^3b^2c-2528a^3bc^2-1664a^3c^3-720a^2b^4-3080a^2b^3c-800a^2b^2c^2-3080a^2bc^3-720a^2c^4+320ab^5+3072ab^4c+7552ab^3c^2+7552ab^2c^3+3072abc^4+320ac^5+432b^6+3132b^5c+8496b^4c^2+12168b^3c^3+8496b^2c^4+3132bc^5+432c^6)x^4-(96(a+b+c))(4a^4-14a^3b-14a^3c-10a^2b^2+50a^2bc-10a^2c^2+28ab^3+41ab^2c+41abc^2+28ac^3-8b^4-37b^3c-18b^2c^2-37bc^3-8c^4)x^5+(576a^4-144a^2bc-6048ab^2c-6048abc^2-576b^4-7056b^3c-7200b^2c^2-7056bc^3-576c^4)x^6+(1728(a+b+c))(ab+ac-b^2-c^2)x^7+5184bcx^8=0\)


\(ca(-a+b-c)(-3a^3-a^2b-9a^2c+3ab^2+6abc-9ac^2+b^3+3b^2c-bc^2-3c^3)(6a^4+2a^3b+36a^3c-7a^2b^2+22a^2bc+60a^2c^2-2ab^3-6ab^2c+22abc^2+36ac^3+b^4-2b^3c-7b^2c^2+2bc^3+6c^4)+(2(a+b+c))(-a+b-c)(-12a^7-4a^6b-45a^6c+26a^5b^2+112a^5bc-117a^5c^2+8a^4b^3+58a^4b^2c+332a^4bc^2-258a^4c^3-16a^3b^4-122a^3b^3c+24a^3b^2c^2+272a^3bc^3-258a^3c^4-4a^2b^5-31a^2b^4c-204a^2b^3c^2+24a^2b^2c^3+332a^2bc^4-117a^2c^5+2ab^6+26ab^5c-31ab^4c^2-122ab^3c^3+58ab^2c^4+112abc^5-45ac^6+2b^6c-4b^5c^2-16b^4c^3+8b^3c^4+26b^2c^5-4bc^6-12c^7)y+(-72a^8-208a^7b-567a^7c+136a^6b^2-462a^6bc-2430a^6c^2+496a^5b^3+895a^5b^2c-966a^5bc^2-5913a^5c^3-40a^4b^4+908a^4b^3c+2796a^4b^2c^2-2108a^4bc^3-7956a^4c^4-368a^3b^5-697a^3b^4c+1748a^3b^3c^2+4538a^3b^2c^3-2108a^3bc^4-5913a^3c^5-40a^2b^6-398a^2b^5c-710a^2b^4c^2+1748a^2b^3c^3+2796a^2b^2c^4-966a^2bc^5-2430a^2c^6+80ab^7+241ab^6c-398ab^5c^2-697ab^4c^3+908ab^3c^4+895ab^2c^5-462abc^6-567ac^7+16b^8+80b^7c-40b^6c^2-368b^5c^3-40b^4c^4+496b^3c^5+136b^2c^6-208bc^7-72c^8)y^2+(4(a+b+c))(-45a^6+221a^5b-276a^5c-6a^4b^2+561a^4bc-579a^4c^2-202a^3b^3+332a^3b^2c+1234a^3bc^2-792a^3c^3-29a^2b^4-494a^2b^3c-292a^2b^2c^2+1234a^2bc^3-579a^2c^4+45ab^5-168ab^4c-494ab^3c^2+332ab^2c^3+561abc^4-276ac^5+16b^6+45b^5c-29b^4c^2-202b^3c^3-6b^2c^4+221bc^5-45c^6)y^3+(432a^6+320a^5b+3132a^5c-720a^4b^2+3072a^4bc+8496a^4c^2-1664a^3b^3-3080a^3b^2c+7552a^3bc^2+12168a^3c^3-368a^2b^4-2528a^2b^3c-800a^2b^2c^2+7552a^2bc^3+8496a^2c^4+1344ab^5+1900ab^4c-2528ab^3c^2-3080ab^2c^3+3072abc^4+3132ac^5+656b^6+1344b^5c-368b^4c^2-1664b^3c^3-720b^2c^4+320bc^5+432c^6)y^4-(96(a+b+c))(-8a^4+28a^3b-37a^3c-10a^2b^2+41a^2bc-18a^2c^2-14ab^3+50ab^2c+41abc^2-37ac^3+4b^4-14b^3c-10b^2c^2+28bc^3-8c^4)y^5+(-576a^4-7056a^3c-6048a^2bc-7200a^2c^2-144ab^2c-6048abc^2-7056ac^3+576b^4-576c^4)y^6+(1728(a+b+c))(-a^2+ab+bc-c^2)y^7+5184cay^8=0\)


\(ab(-b+c-a)(-3a^3-9a^2b-a^2c-9ab^2+6abc+3ac^2-3b^3-b^2c+3bc^2+c^3)(6a^4+36a^3b+2a^3c+60a^2b^2+22a^2bc-7a^2c^2+36ab^3+22ab^2c-6abc^2-2ac^3+6b^4+2b^3c-7b^2c^2-2bc^3+c^4)+(2(a+b+c))(-b+c-a)(-12a^7-45a^6b-4a^6c-117a^5b^2+112a^5bc+26a^5c^2-258a^4b^3+332a^4b^2c+58a^4bc^2+8a^4c^3-258a^3b^4+272a^3b^3c+24a^3b^2c^2-122a^3bc^3-16a^3c^4-117a^2b^5+332a^2b^4c+24a^2b^3c^2-204a^2b^2c^3-31a^2bc^4-4a^2c^5-45ab^6+112ab^5c+58ab^4c^2-122ab^3c^3-31ab^2c^4+26abc^5+2ac^6-12b^7-4b^6c+26b^5c^2+8b^4c^3-16b^3c^4-4b^2c^5+2bc^6)z+(-72a^8-567a^7b-208a^7c-2430a^6b^2-462a^6bc+136a^6c^2-5913a^5b^3-966a^5b^2c+895a^5bc^2+496a^5c^3-7956a^4b^4-2108a^4b^3c+2796a^4b^2c^2+908a^4bc^3-40a^4c^4-5913a^3b^5-2108a^3b^4c+4538a^3b^3c^2+1748a^3b^2c^3-697a^3bc^4-368a^3c^5-2430a^2b^6-966a^2b^5c+2796a^2b^4c^2+1748a^2b^3c^3-710a^2b^2c^4-398a^2bc^5-40a^2c^6-567ab^7-462ab^6c+895ab^5c^2+908ab^4c^3-697ab^3c^4-398ab^2c^5+241abc^6+80ac^7-72b^8-208b^7c+136b^6c^2+496b^5c^3-40b^4c^4-368b^3c^5-40b^2c^6+80bc^7+16c^8)z^2+(4(a+b+c))(-45a^6-276a^5b+221a^5c-579a^4b^2+561a^4bc-6a^4c^2-792a^3b^3+1234a^3b^2c+332a^3bc^2-202a^3c^3-579a^2b^4+1234a^2b^3c-292a^2b^2c^2-494a^2bc^3-29a^2c^4-276ab^5+561ab^4c+332ab^3c^2-494ab^2c^3-168abc^4+45ac^5-45b^6+221b^5c-6b^4c^2-202b^3c^3-29b^2c^4+45bc^5+16c^6)z^3+(432a^6+3132a^5b+320a^5c+8496a^4b^2+3072a^4bc-720a^4c^2+12168a^3b^3+7552a^3b^2c-3080a^3bc^2-1664a^3c^3+8496a^2b^4+7552a^2b^3c-800a^2b^2c^2-2528a^2bc^3-368a^2c^4+3132ab^5+3072ab^4c-3080ab^3c^2-2528ab^2c^3+1900abc^4+1344ac^5+432b^6+320b^5c-720b^4c^2-1664b^3c^3-368b^2c^4+1344bc^5+656c^6)z^4-(96(a+b+c))(-8a^4-37a^3b+28a^3c-18a^2b^2+41a^2bc-10a^2c^2-37ab^3+41ab^2c+50abc^2-14ac^3-8b^4+28b^3c-10b^2c^2-14bc^3+4c^4)z^5+(-576a^4-7056a^3b-7200a^2b^2-6048a^2bc-7056ab^3-6048ab^2c-144abc^2-576b^4+576c^4)z^6+(1728(a+b+c))(-a^2+ac-b^2+bc)z^7+5184abz^8=0\)


毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2020-7-30 11:40:41 | 显示全部楼层
本帖最后由 chyanog 于 2020-7-30 12:15 编辑
数学星空 发表于 2014-2-17 19:13
在9#我已提到,其实不需要太多计算

分别在四边形\(APQB,APQC,BPQC\)中利用11#公式可得


其实这个公式还有更简单的表示形式
令\(t_1=v_1^2-v_2^2-w_1^2+w_2^2\),\(t_2=w_1^2-w_2^2-u_1^2+u_2^2\),\(t_3=u_1^2-u_2^2-v_1^2+v_2^2\),则
\((a-b-c) (a+b-c) (a-b+c) (a+b+c) x^2=a^2 t_2 t_3+b^2 t_3 t_1+c^2 t_1 t_2\)

\(x=\sqrt{\displaystyle{\frac{a^2 t_2 t_3+b^2 t_3 t_1+c^2 t_1 t_2}{(a-b-c) (a+b-c) (a-b+c) (a+b+c)}}}\)
当P、Q两点在三角形外部时也是成立的,可以用下面的Mathematica代码检验

  1. dist = Sqrt[(a^2 t2 t3+b^2 t3 t1+c^2 t1 t2)/((a-b-c) (a+b-c) (a-b+c) (a+b+c))]/. {t1->v1^2-v2^2-w1^2+w2^2, t2->w1^2-w2^2-u1^2+u2^2, t3->u1^2-u2^2-v1^2+v2^2};
  2. dist /. Thread[{a,b,c,u1,v1,w1,u2,v2,w2}->{1,1,1,Sqrt[3]/2,1/2,1/2,Sqrt[3]/3,Sqrt[3]/3,Sqrt[3]/3}]
  3. dist /. Thread[{a,b,c,u1,v1,w1,u2,v2,w2}->{3,4,5,16/5,9/5,12/5,5/2,5/2,5/2}]
复制代码

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经验算一个{a=3,b=4,c=5,u1=4,v1=2,w1=1.012240884,u2=3.5,v2=2.5,w2=4.026049785,x=3.201404028}是正确的,不知道你是如何得到的这个公式?  发表于 2020-8-3 21:10

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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2020-8-13 17:15:41 | 显示全部楼层
本帖最后由 chyanog 于 2020-8-14 08:42 编辑
chyanog 发表于 2020-7-30 11:40
其实这个公式还有更简单的表示形式
令\(t_1=v_1^2-v_2^2-w_1^2+w_2^2\),\(t_2=w_1^2-w_2^2-u_1^2+u_2 ...


先算出PQ的重心坐标,再用两点距离公式 https://en.wikipedia.org/wiki/Ba ... ance_between_points
  1. signArea=Det[{#-#2,#2-#3}]/2&;

  2. gb=GroebnerBasis[{
  3. k signArea[{x1,y1},{x2,y2},{x3,y3}]==signArea[{x2,y2},{x3,y3},{x,y}],
  4. (x-x1)^2+(y-y1)^2==u^2,
  5. (x-x2)^2+(y-y2)^2==v^2,
  6. (x-x3)^2+(y-y3)^2==w^2,

  7. (x2-x3)^2+(y2-y3)^2==a^2,
  8. (x3-x1)^2+(y3-y1)^2==b^2
  9. }/.Thread[{x1,y1,x2,y2}->{0,0,c,0}],{k},{x3,y3,x,y}];

  10. func=Function[{a,b,c,v,w,u},k/.Solve[gb[[-1]]==0,k][[1]]//Evaluate]

  11. P=func@@@{{a,b,c,v1,w1,u1},{b,c,a,w1,u1,v1},{c,a,b,u1,v1,w1}};
  12. Q=func@@@{{a,b,c,v2,w2,u2},{b,c,a,w2,u2,v2},{c,a,b,u2,v2,w2}};
  13. -a^2 y z-b^2 z x-c^2 x y/.Thread[{x,y,z}->P-Q]//Simplify
复制代码
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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