这个三元方程组为什么解不出来？

TSC999

上面 10# 的这个结果是【悠闲数学娱乐论坛(第2版) 】的版主 Kuing 给出的。这位版主精通不等式和几何。

由10# 表达式搞出面积平方的解析式，目前任何软件都无能为力。就像我们能够看到木星，那是客观存在的东西，但是人类目前登陆不上去。

chyanog

刚才无意中发现其实论坛里已经讨论过这个问题了
无法求解的两道难题

Mathematica代码

1. GroebnerBasis[{b c (b+c+a) (b+c-a)==ta^2 (b+c)^2,c a(c+a+b) (c+a-b)==tb^2 (c+a)^2,a b (a+b+c) (a+b-c)==tc^2 (a+b)^2},a,{b,c},
   
2. MonomialOrder->EliminationOrder]//AbsoluteTiming

复制代码

耗时2分钟多

王守恩

已知三角形三个角的平分线长度 ta, tb, tc, 求三条边 a, b, c 的长度。

\(1，不妨设a=\sin(2A)\ \ \ \ b=\sin(2B)\ \ \ \ c=\sin(2C)\)
\(2，根据面积相等，我们有：\)
\( t_a\sin(2A)\sin(2B+A)=t_b\sin(2B)\sin(2A+B)=t_c\sin(2A+2B)\cos(A-B)\)
\(3，解下列方程可得2A，2B，A，B\)
\( \frac{t_a}{t_b}=\frac{\sin(2B)\sin(2A+B)}{\sin(2A)\sin(2B+A)}\)
\( \frac{t_b}{t_c}=\frac{\sin(2A+2B)\cos(A-B)}{\sin(2B)\sin(2A+B)}\)
\(4，解下列方程可得a，b，c\)
\( \frac{a}{\cos(A-B)}=\frac{t_c}{\sin(2B)}\)
\( \frac{b}{\cos(A-B)}=\frac{t_c}{\sin(2A)}\)
\( \frac{c}{\sin(2B+A)}=\frac{t_a}{\sin(2B)}\)


补充内容 (2018-8-5 10:46):
第3条是个 “超越方程”，解是唯一的。解这个 “超越方程” 很难吗？ 我们不妨约定 ta < tb < tc，则   sin(2A) > sin(2B) > sin(2C)。具体的题目，手工找一找，好像也不难。

mathematica

想一想真觉得人类很牛！居然发明了方程思想，
用算术永远解决不了这类问题，
用方程都这么难，而况乎算术。

mathematica

chyanog 发表于 2018-7-27 20:17
刚才无意中发现其实论坛里已经讨论过这个问题了
无法求解的两道难题

我计算了一下，是关于a的24次方程
\[\left\{\left(256 \text{tb}^2 \text{tc}^6 \text{ta}^{12}-512 \text{tb}^4 \text{tc}^4 \text{ta}^{12}+256 \text{tb}^6 \text{tc}^2 \text{ta}^{12}-512 \text{tb}^2 \text{tc}^8 \text{ta}^{10}-512 \text{tb}^8 \text{tc}^2 \text{ta}^{10}+256 \text{tb}^2 \text{tc}^{10} \text{ta}^8+1536 \text{tb}^4 \text{tc}^8 \text{ta}^8+768 \text{tb}^6 \text{tc}^6 \text{ta}^8+1536 \text{tb}^8 \text{tc}^4 \text{ta}^8+256 \text{tb}^{10} \text{tc}^2 \text{ta}^8-1024 \text{tb}^4 \text{tc}^{10} \text{ta}^6-2560 \text{tb}^6 \text{tc}^8 \text{ta}^6-2560 \text{tb}^8 \text{tc}^6 \text{ta}^6-1024 \text{tb}^{10} \text{tc}^4 \text{ta}^6+1536 \text{tb}^6 \text{tc}^{10} \text{ta}^4+2560 \text{tb}^8 \text{tc}^8 \text{ta}^4+1536 \text{tb}^{10} \text{tc}^6 \text{ta}^4-1024 \text{tb}^8 \text{tc}^{10} \text{ta}^2-1024 \text{tb}^{10} \text{tc}^8 \text{ta}^2+256 \text{tb}^{10} \text{tc}^{10}\right) a^{24}+\left(256 \text{tb}^8 \text{ta}^{14}+256 \text{tc}^8 \text{ta}^{14}-512 \text{tb}^4 \text{tc}^4 \text{ta}^{14}-512 \text{tb}^{10} \text{ta}^{12}-512 \text{tc}^{10} \text{ta}^{12}-2304 \text{tb}^2 \text{tc}^8 \text{ta}^{12}+768 \text{tb}^4 \text{tc}^6 \text{ta}^{12}+768 \text{tb}^6 \text{tc}^4 \text{ta}^{12}-2304 \text{tb}^8 \text{tc}^2 \text{ta}^{12}+256 \text{tb}^{12} \text{ta}^{10}+256 \text{tc}^{12} \text{ta}^{10}+2816 \text{tb}^2 \text{tc}^{10} \text{ta}^{10}+10240 \text{tb}^4 \text{tc}^8 \text{ta}^{10}-256 \text{tb}^6 \text{tc}^6 \text{ta}^{10}+10240 \text{tb}^8 \text{tc}^4 \text{ta}^{10}+2816 \text{tb}^{10} \text{tc}^2 \text{ta}^{10}-1536 \text{tb}^2 \text{tc}^{12} \text{ta}^8-7936 \text{tb}^4 \text{tc}^{10} \text{ta}^8-16896 \text{tb}^6 \text{tc}^8 \text{ta}^8-16896 \text{tb}^8 \text{tc}^6 \text{ta}^8-7936 \text{tb}^{10} \text{tc}^4 \text{ta}^8-1536 \text{tb}^{12} \text{tc}^2 \text{ta}^8+3584 \text{tb}^4 \text{tc}^{12} \text{ta}^6+11264 \text{tb}^6 \text{tc}^{10} \text{ta}^6+18432 \text{tb}^8 \text{tc}^8 \text{ta}^6+11264 \text{tb}^{10} \text{tc}^6 \text{ta}^6+3584 \text{tb}^{12} \text{tc}^4 \text{ta}^6-4096 \text{tb}^6 \text{tc}^{12} \text{ta}^4-7936 \text{tb}^8 \text{tc}^{10} \text{ta}^4-7936 \text{tb}^{10} \text{tc}^8 \text{ta}^4-4096 \text{tb}^{12} \text{tc}^6 \text{ta}^4+2304 \text{tb}^8 \text{tc}^{12} \text{ta}^2+2816 \text{tb}^{10} \text{tc}^{10} \text{ta}^2+2304 \text{tb}^{12} \text{tc}^8 \text{ta}^2-512 \text{tb}^{10} \text{tc}^{12}-512 \text{tb}^{12} \text{tc}^{10}\right) a^{22}+\left(-1280 \text{tb}^{10} \text{ta}^{14}-1280 \text{tc}^{10} \text{ta}^{14}-1280 \text{tb}^2 \text{tc}^8 \text{ta}^{14}+2560 \text{tb}^4 \text{tc}^6 \text{ta}^{14}+2560 \text{tb}^6 \text{tc}^4 \text{ta}^{14}-1280 \text{tb}^8 \text{tc}^2 \text{ta}^{14}+768 \text{tb}^{12} \text{ta}^{12}+768 \text{tc}^{12} \text{ta}^{12}+9568 \text{tb}^2 \text{tc}^{10} \text{ta}^{12}+10240 \text{tb}^4 \text{tc}^8 \text{ta}^{12}-5312 \text{tb}^6 \text{tc}^6 \text{ta}^{12}+10240 \text{tb}^8 \text{tc}^4 \text{ta}^{12}+9568 \text{tb}^{10} \text{tc}^2 \text{ta}^{12}-512 \text{tb}^{14} \text{ta}^{10}-512 \text{tc}^{14} \text{ta}^{10}-4384 \text{tb}^2 \text{tc}^{12} \text{ta}^{10}-25440 \text{tb}^4 \text{tc}^{10} \text{ta}^{10}-31072 \text{tb}^6 \text{tc}^8 \text{ta}^{10}-31072 \text{tb}^8 \text{tc}^6 \text{ta}^{10}-25440 \text{tb}^{10} \text{tc}^4 \text{ta}^{10}-4384 \text{tb}^{12} \text{tc}^2 \text{ta}^{10}+2304 \text{tb}^2 \text{tc}^{14} \text{ta}^8+11232 \text{tb}^4 \text{tc}^{12} \text{ta}^8+35104 \text{tb}^6 \text{tc}^{10} \text{ta}^8+36960 \text{tb}^8 \text{tc}^8 \text{ta}^8+35104 \text{tb}^{10} \text{tc}^6 \text{ta}^8+11232 \text{tb}^{12} \text{tc}^4 \text{ta}^8+2304 \text{tb}^{14} \text{tc}^2 \text{ta}^8-4096 \text{tb}^4 \text{tc}^{14} \text{ta}^6-12480 \text{tb}^6 \text{tc}^{12} \text{ta}^6-24384 \text{tb}^8 \text{tc}^{10} \text{ta}^6-24384 \text{tb}^{10} \text{tc}^8 \text{ta}^6-12480 \text{tb}^{12} \text{tc}^6 \text{ta}^6-4096 \text{tb}^{14} \text{tc}^4 \text{ta}^6+3584 \text{tb}^6 \text{tc}^{14} \text{ta}^4+5696 \text{tb}^8 \text{tc}^{12} \text{ta}^4+8768 \text{tb}^{10} \text{tc}^{10} \text{ta}^4+5696 \text{tb}^{12} \text{tc}^8 \text{ta}^4+3584 \text{tb}^{14} \text{tc}^6 \text{ta}^4-1536 \text{tb}^8 \text{tc}^{14} \text{ta}^2-1568 \text{tb}^{10} \text{tc}^{12} \text{ta}^2-1568 \text{tb}^{12} \text{tc}^{10} \text{ta}^2-1536 \text{tb}^{14} \text{tc}^8 \text{ta}^2+256 \text{tb}^{10} \text{tc}^{14}+736 \text{tb}^{12} \text{tc}^{12}+256 \text{tb}^{14} \text{tc}^{10}\right) a^{20}+\left(256 \text{ta}^{10} \text{tb}^{16}+256 \text{ta}^2 \text{tc}^8 \text{tb}^{16}-1024 \text{ta}^4 \text{tc}^6 \text{tb}^{16}+1536 \text{ta}^6 \text{tc}^4 \text{tb}^{16}-1024 \text{ta}^8 \text{tc}^2 \text{tb}^{16}-32 \text{ta}^{12} \text{tb}^{14}-288 \text{tc}^{12} \text{tb}^{14}+448 \text{ta}^2 \text{tc}^{10} \text{tb}^{14}-992 \text{ta}^4 \text{tc}^8 \text{tb}^{14}+4480 \text{ta}^6 \text{tc}^6 \text{tb}^{14}-6368 \text{ta}^8 \text{tc}^4 \text{tb}^{14}+2752 \text{ta}^{10} \text{tc}^2 \text{tb}^{14}+1888 \text{ta}^{14} \text{tb}^{12}-288 \text{tc}^{14} \text{tb}^{12}-1392 \text{ta}^2 \text{tc}^{12} \text{tb}^{12}-5840 \text{ta}^4 \text{tc}^{10} \text{tb}^{12}+13072 \text{ta}^6 \text{tc}^8 \text{tb}^{12}-28352 \text{ta}^8 \text{tc}^6 \text{tb}^{12}+24672 \text{ta}^{10} \text{tc}^4 \text{tb}^{12}-12912 \text{ta}^{12} \text{tc}^2 \text{tb}^{12}+448 \text{ta}^2 \text{tc}^{14} \text{tb}^{10}-5840 \text{ta}^4 \text{tc}^{12} \text{tb}^{10}+16640 \text{ta}^6 \text{tc}^{10} \text{tb}^{10}-37552 \text{ta}^8 \text{tc}^8 \text{tb}^{10}+49920 \text{ta}^{10} \text{tc}^6 \text{tb}^{10}-32016 \text{ta}^{12} \text{tc}^4 \text{tb}^{10}+6848 \text{ta}^{14} \text{tc}^2 \text{tb}^{10}+256 \text{ta}^2 \text{tc}^{16} \text{tb}^8-992 \text{ta}^4 \text{tc}^{14} \text{tb}^8+13072 \text{ta}^6 \text{tc}^{12} \text{tb}^8-37552 \text{ta}^8 \text{tc}^{10} \text{tb}^8+56528 \text{ta}^{10} \text{tc}^8 \text{tb}^8-16352 \text{ta}^{12} \text{tc}^6 \text{tb}^8-1888 \text{ta}^{14} \text{tc}^4 \text{tb}^8-1024 \text{ta}^4 \text{tc}^{16} \text{tb}^6+4480 \text{ta}^6 \text{tc}^{14} \text{tb}^6-28352 \text{ta}^8 \text{tc}^{12} \text{tb}^6+49920 \text{ta}^{10} \text{tc}^{10} \text{tb}^6-16352 \text{ta}^{12} \text{tc}^8 \text{tb}^6-13696 \text{ta}^{14} \text{tc}^6 \text{tb}^6+1536 \text{ta}^6 \text{tc}^{16} \text{tb}^4-6368 \text{ta}^8 \text{tc}^{14} \text{tb}^4+24672 \text{ta}^{10} \text{tc}^{12} \text{tb}^4-32016 \text{ta}^{12} \text{tc}^{10} \text{tb}^4-1888 \text{ta}^{14} \text{tc}^8 \text{tb}^4-1024 \text{ta}^8 \text{tc}^{16} \text{tb}^2+2752 \text{ta}^{10} \text{tc}^{14} \text{tb}^2-12912 \text{ta}^{12} \text{tc}^{12} \text{tb}^2+6848 \text{ta}^{14} \text{tc}^{10} \text{tb}^2+256 \text{ta}^{10} \text{tc}^{16}-32 \text{ta}^{12} \text{tc}^{14}+1888 \text{ta}^{14} \text{tc}^{12}\right) a^{18}+\left(-288 \text{ta}^{12} \text{tb}^{16}-288 \text{ta}^2 \text{tc}^{10} \text{tb}^{16}-288 \text{ta}^4 \text{tc}^8 \text{tb}^{16}+576 \text{ta}^6 \text{tc}^6 \text{tb}^{16}+576 \text{ta}^8 \text{tc}^4 \text{tb}^{16}-288 \text{ta}^{10} \text{tc}^2 \text{tb}^{16}-720 \text{ta}^{14} \text{tb}^{14}+81 \text{tc}^{14} \text{tb}^{14}+966 \text{ta}^2 \text{tc}^{12} \text{tb}^{14}+143 \text{ta}^4 \text{tc}^{10} \text{tb}^{14}-2140 \text{ta}^6 \text{tc}^8 \text{tb}^{14}+5967 \text{ta}^8 \text{tc}^6 \text{tb}^{14}-8666 \text{ta}^{10} \text{tc}^4 \text{tb}^{14}+3217 \text{ta}^{12} \text{tc}^2 \text{tb}^{14}+966 \text{ta}^2 \text{tc}^{14} \text{tb}^{12}+6480 \text{ta}^4 \text{tc}^{12} \text{tb}^{12}+3050 \text{ta}^6 \text{tc}^{10} \text{tb}^{12}+20854 \text{ta}^8 \text{tc}^8 \text{tb}^{12}-18872 \text{ta}^{10} \text{tc}^6 \text{tb}^{12}+35778 \text{ta}^{12} \text{tc}^4 \text{tb}^{12}-10608 \text{ta}^{14} \text{tc}^2 \text{tb}^{12}-288 \text{ta}^2 \text{tc}^{16} \text{tb}^{10}+143 \text{ta}^4 \text{tc}^{14} \text{tb}^{10}+3050 \text{ta}^6 \text{tc}^{12} \text{tb}^{10}+19615 \text{ta}^8 \text{tc}^{10} \text{tb}^{10}-34894 \text{ta}^{10} \text{tc}^8 \text{tb}^{10}+43775 \text{ta}^{12} \text{tc}^6 \text{tb}^{10}-11216 \text{ta}^{14} \text{tc}^4 \text{tb}^{10}-288 \text{ta}^4 \text{tc}^{16} \text{tb}^8-2140 \text{ta}^6 \text{tc}^{14} \text{tb}^8+20854 \text{ta}^8 \text{tc}^{12} \text{tb}^8-34894 \text{ta}^{10} \text{tc}^{10} \text{tb}^8+37340 \text{ta}^{12} \text{tc}^8 \text{tb}^8+22544 \text{ta}^{14} \text{tc}^6 \text{tb}^8+576 \text{ta}^6 \text{tc}^{16} \text{tb}^6+5967 \text{ta}^8 \text{tc}^{14} \text{tb}^6-18872 \text{ta}^{10} \text{tc}^{12} \text{tb}^6+43775 \text{ta}^{12} \text{tc}^{10} \text{tb}^6+22544 \text{ta}^{14} \text{tc}^8 \text{tb}^6+576 \text{ta}^8 \text{tc}^{16} \text{tb}^4-8666 \text{ta}^{10} \text{tc}^{14} \text{tb}^4+35778 \text{ta}^{12} \text{tc}^{12} \text{tb}^4-11216 \text{ta}^{14} \text{tc}^{10} \text{tb}^4-288 \text{ta}^{10} \text{tc}^{16} \text{tb}^2+3217 \text{ta}^{12} \text{tc}^{14} \text{tb}^2-10608 \text{ta}^{14} \text{tc}^{12} \text{tb}^2-288 \text{ta}^{12} \text{tc}^{16}-720 \text{ta}^{14} \text{tc}^{14}\right) a^{16}+\left(81 \text{ta}^{14} \text{tb}^{16}+81 \text{ta}^2 \text{tc}^{12} \text{tb}^{16}+1254 \text{ta}^4 \text{tc}^{10} \text{tb}^{16}+1215 \text{ta}^6 \text{tc}^8 \text{tb}^{16}+84 \text{ta}^8 \text{tc}^6 \text{tb}^{16}+1215 \text{ta}^{10} \text{tc}^4 \text{tb}^{16}+1254 \text{ta}^{12} \text{tc}^2 \text{tb}^{16}-432 \text{ta}^2 \text{tc}^{14} \text{tb}^{14}-1868 \text{ta}^4 \text{tc}^{12} \text{tb}^{14}-860 \text{ta}^6 \text{tc}^{10} \text{tb}^{14}-2152 \text{ta}^8 \text{tc}^8 \text{tb}^{14}+1672 \text{ta}^{10} \text{tc}^6 \text{tb}^{14}-8012 \text{ta}^{12} \text{tc}^4 \text{tb}^{14}+3972 \text{ta}^{14} \text{tc}^2 \text{tb}^{14}+81 \text{ta}^2 \text{tc}^{16} \text{tb}^{12}-1868 \text{ta}^4 \text{tc}^{14} \text{tb}^{12}-12772 \text{ta}^6 \text{tc}^{12} \text{tb}^{12}-10400 \text{ta}^8 \text{tc}^{10} \text{tb}^{12}-11013 \text{ta}^{10} \text{tc}^8 \text{tb}^{12}-34520 \text{ta}^{12} \text{tc}^6 \text{tb}^{12}+22468 \text{ta}^{14} \text{tc}^4 \text{tb}^{12}+1254 \text{ta}^4 \text{tc}^{16} \text{tb}^{10}-860 \text{ta}^6 \text{tc}^{14} \text{tb}^{10}-10400 \text{ta}^8 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\text{tb}^{16}-3808 \text{ta}^{10} \text{tc}^6 \text{tb}^{16}-2672 \text{ta}^{12} \text{tc}^4 \text{tb}^{16}-432 \text{ta}^{14} \text{tc}^2 \text{tb}^{16}+945 \text{ta}^4 \text{tc}^{14} \text{tb}^{14}+2740 \text{ta}^6 \text{tc}^{12} \text{tb}^{14}-442 \text{ta}^8 \text{tc}^{10} \text{tb}^{14}+3748 \text{ta}^{10} \text{tc}^8 \text{tb}^{14}+4049 \text{ta}^{12} \text{tc}^6 \text{tb}^{14}-8336 \text{ta}^{14} \text{tc}^4 \text{tb}^{14}-432 \text{ta}^4 \text{tc}^{16} \text{tb}^{12}+2740 \text{ta}^6 \text{tc}^{14} \text{tb}^{12}+9980 \text{ta}^8 \text{tc}^{12} \text{tb}^{12}+17532 \text{ta}^{10} \text{tc}^{10} \text{tb}^{12}+11252 \text{ta}^{12} \text{tc}^8 \text{tb}^{12}-19760 \text{ta}^{14} \text{tc}^6 \text{tb}^{12}-2672 \text{ta}^6 \text{tc}^{16} \text{tb}^{10}-442 \text{ta}^8 \text{tc}^{14} \text{tb}^{10}+17532 \text{ta}^{10} \text{tc}^{12} \text{tb}^{10}+4966 \text{ta}^{12} \text{tc}^{10} \text{tb}^{10}+28528 \text{ta}^{14} \text{tc}^8 \text{tb}^{10}-3808 \text{ta}^8 \text{tc}^{16} 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\text{tc}^{14} \text{tb}^{12}-5888 \text{ta}^{10} \text{tc}^{12} \text{tb}^{12}+3760 \text{ta}^{12} \text{tc}^{10} \text{tb}^{12}+3232 \text{ta}^{14} \text{tc}^8 \text{tb}^{12}+3456 \text{ta}^8 \text{tc}^{16} \text{tb}^{10}+48 \text{ta}^{10} \text{tc}^{14} \text{tb}^{10}+3760 \text{ta}^{12} \text{tc}^{12} \text{tb}^{10}-23936 \text{ta}^{14} \text{tc}^{10} \text{tb}^{10}+5184 \text{ta}^{10} \text{tc}^{16} \text{tb}^8+1936 \text{ta}^{12} \text{tc}^{14} \text{tb}^8+3232 \text{ta}^{14} \text{tc}^{12} \text{tb}^8+3456 \text{ta}^{12} \text{tc}^{16} \text{tb}^6+7872 \text{ta}^{14} \text{tc}^{14} \text{tb}^6+864 \text{ta}^{14} \text{tc}^{16} \text{tb}^4\right) a^{10}+\left(-768 \text{ta}^8 \text{tc}^{12} \text{tb}^{16}-2304 \text{ta}^{10} \text{tc}^{10} \text{tb}^{16}-2304 \text{ta}^{12} \text{tc}^8 \text{tb}^{16}-768 \text{ta}^{14} \text{tc}^6 \text{tb}^{16}+1120 \text{ta}^8 \text{tc}^{14} \text{tb}^{14}+1472 \text{ta}^{10} \text{tc}^{12} \text{tb}^{14}-928 \text{ta}^{12} \text{tc}^{10} 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\text{tb}^{10}+256 \text{ta}^{14} \text{tc}^{16} \text{tb}^8\right) a^6+256 \text{ta}^{12} \text{tb}^{14} \text{tc}^{14} a^4\right\}\]

mathematica

chyanog 发表于 2018-7-27 20:17
刚才无意中发现其实论坛里已经讨论过这个问题了
无法求解的两道难题

1. Clear["Global`*"];(*mathematica11.2,win7(64bit)Clear all variables*)
   
2. f1=tc^2*(a+b)^2-(a*b*((a+b)^2-c^2))
   
3. f2=tb^2*(a+c)^2-(a*c*((a+c)^2-b^2))
   
4. f3=ta^2*(b+c)^2-(c*b*((c+b)^2-a^2))
   
5. g1=Resultant[f1,f2,b]
   
6. g2=Resultant[f2,f3,b]
   
7. h1=Resultant[g1,g2,c]
   
8. Collect[h1,a]
   

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为什么我用结式最后得到的是关于a的32次方程，除以2，也是16次方程，
而你的是24次方程，(24-4)/2=10次方程
这让我困惑了！