关于三角形的几个问题

数学星空

关于27#的问题，即马尔法蒂问题（Malfatti's Problem）：在一个已知三角形内画三个圆，每个圆与其他两个圆以及三角形的两边相切。
容易得到
\( r_2 \cot(B/2)+2\sqrt{r_2r_3}+r_3 \cot(C/2)=a \)
\( r_1 \cot(A/2)+2\sqrt{r_1r_3}+r_3 \cot(C/2)=b\)
\( r_1 \cot(A/2)+2\sqrt{r_1r_2}+r_2 \cot(B/2)=c \)
即
\( r_2s(s-b)+2S\sqrt{r_2r_3}+r_3s(s-c)=aS \)
\( r_1s(s-a)+2S\sqrt{r_1r_3}+r_3s(s-c)=bS \)
\( r_1s(s-a)+2S\sqrt{r_1r_2}+r_2s(s-b)=cS\)
消元得到：
\( (-b-c+a)(b-c+a)^6(-b+c+a)^6+16(b+c+a)(-b-c+a)(b-c+a)^5(-b+c+a)^5r_1+128(a^5-a^3b^2-3a^3bc-a^3c^2-a^2b^3+6a^2b^2c+6a^2bc^2-a^2c^3-3ab^3c+6ab^2c^2-3abc^3+b^5-b^3c^2-b^2c^3+c^5)(b-c+a)^3(-b+c+a)^3r_1^2+128(-b-c+a)(5a^3+3a^2b+3a^2c+3ab^2-14abc+3ac^2+5b^3+3b^2c+3bc^2+5c^3)(b-c+a)^3(-b+c+a)^3r_1^3+128(-b-c+a)(17a^4-2a^2b^2-2a^2c^2+17b^4-2b^2c^2+17c^4)(b-c+a)^2(-b+c+a)^2r_1^4+1024(5a^2+2ab+2ac+5b^2+2bc+5c^2)(-b-c+a)^2(b-c+a)^2(-b+c+a)^2r_1^5+8192(b-c+a)(-b+c+a)(a^3+abc+b^3+c^3)(-b-c+a)^2r_1^6+8192(b-c+a)(b+c+a)(-b+c+a)(-b-c+a)^3r_1^7+4096(b+c+a)^2(-b-c+a)^3r_1^8＝0 \)

\((-b+c+a)(-b-c+a)^6(b-c+a)^6-16(b+c+a)(-b+c+a)(-b-c+a)^5(b-c+a)^5r_2+128(a^5-a^3b^2-3a^3bc-a^3c^2-a^2b^3+6a^2b^2c+6a^2bc^2-a^2c^3-3ab^3c+6ab^2c^2-3abc^3+b^5-b^3c^2-b^2c^3+c^5)(-b-c+a)^3(b-c+a)^2r_2^2-128(-b+c+a)(5a^3+3a^2b+3a^2c+3ab^2-14abc+3ac^2+5b^3+3b^2c+3bc^2+5c^3)(-b-c+a)^3(b-c+a)^3r_2^3+128(-b+c+a)(17a^4-2a^2b^2-2a^2c^2+17b^4-2b^2c^2+17c^4)(-b-c+a)^2(b-c+a)^2r_2^4-1024(5a^2+2ab+2ac+5b^2+2bc+5c^2)(-b-c+a)^2(b-c+a)^2(-b+c+a)^2r_2^5+8192(b-c+a)(-b-c+a)(a^3+abc+b^3+c^3)(-b+c+a)^2r_2^6-8192(b-c+a)(b+c+a)(-b-c+a)(-b+c+a)^3r_2^7+4096(b+c+a)^2(-b+c+a)^3r_2^8＝0 \)

\( (b-c+a)(-b-c+a)^6(-b+c+a)^6-16(b+c+a)(b-c+a)(-b-c+a)^5(-b+c+a)^5r_3+128(a^5-a^3b^2-3a^3bc-a^3c^2-a^2b^3+6a^2b^2c+6a^2bc^2-a^2c^3-3ab^3c+6ab^2c^2-3abc^3+b^5-b^3c^2-b^2c^3+c^5)(-b-c+a)^3(-b+c+a)^3r_3^2-128(b-c+a)(5a^3+3a^2b+3a^2c+3ab^2-14abc+3ac^2+5b^3+3b^2c+3bc^2+5c^3)(-b-c+a)^3(-b+c+a)^3r_3^3+128(b-c+a)(17a^4-2a^2b^2-2a^2c^2+17b^4-2b^2c^2+17c^4)(-b-c+a)^2(-b+c+a)^2r_3^4-1024(5a^2+2ab+2ac+5b^2+2bc+5c^2)(-b-c+a)^2(b-c+a)^2(-b+c+a)^2r_3^5+8192(-b-c+a)(-b+c+a)(a^3+abc+b^3+c^3)(b-c+a)^2r_3^6-8192(b+c+a)(-b+c+a)(-b-c+a)(b-c+a)^3r_3^7+4096(b+c+a)^2(b-c+a)^3r_3^8＝0 \)

注： \( r_1 \) 取第三个正实根（根从小到大排列）， \( r_2 \) 取第二个正实根（根从小到大排列）， \( r_3 \) 取第一个正实根（根从小到大排列）
例：
\( a = 3, b = 4, c = 5, r_1 = 0.7519993534, r_2 = 0.6648941885, r_3 =0 .5079339624 \)
\( a = 3, b = 4, c = 6, r_1 =0.6660007375, r_2 = 0.6161503060, r_3 = 0.3399085088 \)
\( a = 4, b = 5, c = 6, r_1 =0 .9619865980, r_2 = 0.8635452050, r_3 = 0.7086379575 \)

数学星空

若 \( \triangle ABC \) 内接四个等圆，且有三个圆分别相切于三角形角 \(A,B,C \) 的两条边，且第四个圆相切于三个圆，求此等圆的半径 \( r_0 \) ？(三角形三边为 \( a,b,c \) )

此问题来自于http://bbs.cnool.net/cthread-104971475.html

设分别相切于角A,B,C的两边的圆心分别为 \( O_1，O_2 ，O_3, \) 第四个圆心为 \( O_4 \) (\( \odot O_4\) 相切于 \( \odot O_1, \odot O_2,\odot O_3 \) ),
且 \( O_1O_2=z,O_1O_3=y,O_2O_3=x \)

\[ \begin{equation}\left\{ \eqalign{ x=a-r_0\left( \cot\frac B2+\cot\frac C2 \right)=a\left( 1-\frac{s\*r_0}{S} \right) \\
y=b-r_0\left( \cot\frac C2+\cot\frac A2 \right)=b\left( 1-\frac{s\*r_0}{S} \right) \\
z=c-r_0\left( \cot\frac A2+\cot\frac B2 \right)=c\left( 1-\frac{s\*r_0}{S} \right) } \right. \end{equation} \]

又由于 \( O_4 \) 为 \( \triangle O_1O_2O_3 \) 的外心，且 \( R_0=2r_0 \)

即 \[ \begin{equation} 2r_0=\frac{xyz}{\sqrt{2x^2y^2+2x^2z^2+2y^2z^2-x^4-y^4-z^4}} \end{equation} \]

由（1~2）得到

\[ r_0=\frac{abc\sqrt{2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4}}{2(2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4+abc(a+b+c))}=\frac{4Sabc}{2(16S^2+2sabc)}=\frac{RS}{2S+sR}=\frac{Rr}{2r+R}=\frac{R}{2+\frac{R}{r}} \]

下面的图片取自于链接网页3#

数学星空

若 点\( P \) 为 \( \triangle ABC \)(三边长分别为 \( a,b,c \) ) 内一点，且 \( \triangle BPC,\triangle APC,\triangle APB \) 的内切圆圆心和半径分别为 \( \odot O_1,\odot O_2,\odot O_3 \) 和 \( r_1,r_2,r_3 \) ,
求 \( S_n=\max(r_1^n+r_2^n+r_3^n) \)及 \( L_n=\min(r_1^n+r_2^n+r_3^n) \)？
注，我们可以先讨论 \( n=1,2 \) ?

数学星空

可以发现：当 \( P \) 点位于 \( \triangle ABC \) 的任一个顶点时 \( S_n=\max(r_1^n+r_2^n+r_3^n)=r^n=(\frac{S}{s})^n
\) 看来只需要讨论 \( L_n=\min(r_1^n+r_2^n+r_3^n)\) ? 猜测仅当 \( r_1=r_2=r_3 \) 时成立??

数学星空

当 \( P \) 点在 \( \triangle ABC  \) 的三边时，相当于求两个圆 \( \odot O_1,\odot O_2 \) 的情形

A1.当两圆均相切于 \( BC \) 边时

\( (-8a^4+16a^3b+16a^3c-32a^2bc-16ab^3+16ab^2c+16abc^2-16ac^3+8b^4-16b^2c^2+8c^4)r_2r_1+6a^4bc-4a^3b^2c-4a^3bc^2+c^6+b^6-2bc^5+4b^3c^3-b^2c^4-2b^5c-b^4c^2-2ac^5+(4a^4-8a^2b^2-8a^2c^2+64a^2r_2^2+4b^4-8b^2c^2+4c^4)r_1^2+(4a^4-8a^2b^2-8a^2c^2+4b^4-8b^2c^2+4c^4)r_2^2+a^6-2ab^5-a^2c^4+4a^3c^3-a^2b^4+4a^3b^3-2a^5b-2a^5c-a^4b^2-a^4c^2+6abc^4-4ab^2c^3-4ab^3c^2+6ab^4c-4a^2bc^3-4a^2b^3c+10a^2b^2c^2＝0 \)

A2.当两圆均相切于 \( AC \) 边时

  \( 6a^4bc-4a^3b^2c-4a^3bc^2+c^6+b^6+(8a^4-16a^3b+16a^2bc-16a^2c^2+16ab^3-32ab^2c+16abc^2-8b^4+16b^3c-16bc^3+8c^4)r_2r_1-2bc^5+4b^3c^3-b^2c^4-2b^5c-b^4c^2-2ac^5+(4a^4-8a^2b^2-8a^2c^2+4b^4-8b^2c^2+4c^4)r_2^2+a^6-2ab^5-a^2c^4+4a^3c^3-a^2b^4+4a^3b^3-2a^5b-2a^5c-a^4b^2-a^4c^2+6abc^4-4ab^2c^3-4ab^3c^2+6ab^4c-4a^2bc^3-4a^2b^3c+10a^2b^2c^2+(4a^4-8a^2b^2-8a^2c^2+4b^4-8b^2c^2+64b^2r_2^2+4c^4)r_1^2=0 \)

A3.当两圆均相切于 \( AB \) 边时

\( 6a^4bc-4a^3b^2c-4a^3bc^2+c^6+(4a^4-8a^2b^2-8a^2c^2+4b^4-8b^2c^2+4c^4+64c^2r_2^2)r_1^2+b^6-2bc^5+4b^3c^3-b^2c^4-2b^5c-b^4c^2-2ac^5+(4a^4-8a^2b^2-8a^2c^2+4b^4-8b^2c^2+4c^4)r_2^2+a^6-2ab^5-a^2c^4+4a^3c^3-a^2b^4+4a^3b^3-2a^5b-2a^5c-a^4b^2-a^4c^2+(8a^4-16a^3c-16a^2b^2+16a^2bc+16ab^2c-32abc^2+16ac^3+8b^4-16b^3c+16bc^3-8c^4)r_2r_1+6abc^4-4ab^2c^3-4ab^3c^2+6ab^4c-4a^2bc^3-4a^2b^3c+10a^2b^2c^2=0 \)

当 \( r_1=r_2=r_0 \) 时
      
B1.当两等圆均相切于 \( BC \) 边时
\(64a^2r_0^4+16(b+c)(b-c+a)(-c+a-b)(a-b+c)r_0^2+(b-c+a)^2(-c+a-b)^2(a-b+c)^2＝0 \)

B2.当两等圆均相切于 \( AC \) 边时
\( 64b^2r_0^4+16(a+c)(b-c+a)(-c+a-b)(a-b+c)r_0^2+(b-c+a)^2(-c+a-b)^2(a-b+c)^2=0 \)

B3.当两等圆均相切于 \( AB \) 边时
\( 64c^2r_0^4+16(a+b)(b-c+a)(-c+a-b)(a-b+c)r_0^2+(b-c+a)^2(-c+a-b)^2(a-b+c)^2=0 \)

数学星空

关于本贴的第二问：
通过相当长时间的消元计算得到，具体结果见附件，有谁有兴趣试着分解一下，我
觉得答案可以简洁很多。
001 - 副本.rar (558.95 KB, 下载次数: 12)

2014-2-14 22:07 上传

点击文件名下载附件

数学星空

楼上的结果由下列方程消元得到：

\(  4(x+y+c)r^2-(x+y-c)(-x+y+c)(x-y+c)＝0 \)
\( 4(x+z+b)r^2-(x+z-b)(-x+z+b)(x-z+b)＝0 \)
\( 4(y+z+a)r^2-(y+z-a)(-y+z+a)(y-z+a)＝0 \)
\( 4r^2(2x+2y+2z+a+b+c)^2+a^4-2a^2b^2-2a^2c^2+b^4-2b^2c^2+c^4＝0 \)

经过分解楼上的结果得到了如下简洁的答案：

\((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 \)

数学星空

现举例计算来验证楼上的结论的是否正确(与23#计算结果对比即可）
1. 取   \( a = 5, b = 12, c = 8 \)
  \(78275778969600r^{16}-25088032663142400r^{14}+2026054624326451200r^{12}-26077679761320345600r^{10}+100412030199497241600r^8-135768794249894400000r^6+81948404811881250000r^4-22739654569921875000r^2+2364627008056640625＝0 \)
取最小正实根 \( r =0.557928817455220\)

2.取 \( a = 5, b = 12, c = 9\)
\(99067782758400r^{16}-35002884487643136r^{14}+3139645196732465152r^{12}-71948685392562094080r^{10}+493460529030713311232r^8-1212323906176294060032r^6+1335383159057948868608r^4-677148573597475799040r^2+128627308957328736256=0\)
取最小正实根得\(r=0.745703010559392\)

3.取 \( a = 5, b = 12, c = 10\)
\(122305904640000r^{16}-47148034424832000r^{14}+4648204904354021376r^{12}-143102337111313219584r^{10}+1298972942870607922176r^8-4275658653825212688384r^6+6325482740192497077840r^4-4310784974826787654008r^2+1100181539196023252409＝0\)
取最小正实根得\(r=0.859897963713285\)

计算结果正确

数学星空

关于在东论上‘天下无毒史’给出的一个图：



并不具备本题一般特性的说明：

A. 若三角形一内点\(P\)与各顶点构成的\(\triangle APC,\triangle APB,\triangle BPC\)的内切圆均两两相切于外公切线\(AP,BP,CP\)，且切于同一点? (并不正确)

我们现在做的就是若A成立，能求出三个内切圆的半径\(r_1,r_2,r_3\) 吗？

其实我们可设\(PA=t+x,PB=t+y,PC=t+z\),则有结论：

\(x=\frac{b+c-a}{2},y=\frac{a+c-b}{2},z=\frac{b+c-a}{2}\)

\(r_1=\sqrt{\frac{yzt}{y+z+t}},r_2=\sqrt{\frac{xzt}{x+z+t}},r_3=\sqrt{\frac{xyt}{x+y+t}}\)

很容易得到如下结果：

\((b+a-c)^2(a+c-b)^2(-b-c+a)^2-(4(b+a-c))(a+c-b)(-b-c+a)(a^2-2ab-2ac+b^2-2bc+c^2)t+(20a^4-16a^3b-16a^3c-8a^2b^2+16a^2bc-8a^2c^2-16ab^3+16ab^2c+16abc^2-16ac^3+20b^4-16b^3c-8b^2c^2-16bc^3+20c^4)t^2=0\)

\((-b-c+a)^2(b+a-c)^4(a+c-b)^4+16(5a^3-3a^2b-3a^2c-ab^2+2abc-ac^2-b^3+b^2c+bc^2-c^3)^2r_1^4-(8(-b-c+a))(3a^3-5a^2b-5a^2c+ab^2-2abc+ac^2+b^3-b^2c-bc^2+c^3)(b+a-c)^2(a+c-b)^2r_1^2=0\)

\((a+c-b)^2(-b-c+a)^4(b+a-c)^4+16(a^3+a^2b-a^2c+3ab^2-2abc-ac^2-5b^3+3b^2c+bc^2+c^3)^2r_2^4+(8(a+c-b))(a^3+a^2b-a^2c-5ab^2-2abc-ac^2+3b^3-5b^2c+bc^2+c^3)(-b-c+a)^2(b+a-c)^2r_2^2=0\)

\((b+a-c)^2(-b-c+a)^4(a+c-b)^4+16(a^3-a^2b+a^2c-ab^2-2abc+3ac^2+b^3+b^2c+3bc^2-5c^3)^2r_3^4+(8(b+a-c))(a^3-a^2b+a^2c-ab^2-2abc-5ac^2+b^3+b^2c-5bc^2+3c^3)(-b-c+a)^2(a+c-b)^2r_3^2=0\)

数学星空

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

中得到了：求三角形内点与各个顶点构成的三个三角形，且三个三角形有等内切圆的最终结果

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



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



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