这个三元方程组为什么解不出来？
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。Clear["Global`*"];
Simplify/(a + b),
tb == Sqrt/(a + c),
ta == Sqrt/(c + b), a > 0, b > 0, c > 0,
a + b > c, b + c > a, c + a > b, Element[{ta, tb, tc}, Reals]}, {a,
b, c}]]
上面这个程序，为什么运行老半天，还不见动静？ 估计是软件黔驴技穷了？ 还是我这程序有问题？还是原问题是不可解的？ 本帖最后由 chyanog 于 2018-7-24 21:36 编辑
因为方程组的解确实比较复杂，你可以先带入具体的数值再解（比如ta->3, tb->4, tc->5），这样解起来就快了，然后RootReduce，观察一下根的形式，可以看出他们是20次方程的一个根，也是10次方程的根的算术平方根。
令三个角的平分线长度分别为x,y,z，那么其中一条边a是下面这个20次方程的一个正实根：
256 a^20 (x-y)^2 y^2 (x+y)^2 (x-z)^2 z^2 (x+z)^2 (x y-x z-y z) (x y+x z-y z) (x y-x z+y z) (x y+x z+y z)+256 a^18 (x^14 y^8-2 x^12 y^10+x^10 y^12-9 x^12 y^8 z^2+11 x^10 y^10 z^2-6 x^8 y^12 z^2-2 x^14 y^4 z^4+3 x^12 y^6 z^4+40 x^10 y^8 z^4-31 x^8 y^10 z^4+14 x^6 y^12 z^4+3 x^12 y^4 z^6-x^10 y^6 z^6-66 x^8 y^8 z^6+44 x^6 y^10 z^6-16 x^4 y^12 z^6+x^14 z^8-9 x^12 y^2 z^8+40 x^10 y^4 z^8-66 x^8 y^6 z^8+72 x^6 y^8 z^8-31 x^4 y^10 z^8+9 x^2 y^12 z^8-2 x^12 z^10+11 x^10 y^2 z^10-31 x^8 y^4 z^10+44 x^6 y^6 z^10-31 x^4 y^8 z^10+11 x^2 y^10 z^10-2 y^12 z^10+x^10 z^12-6 x^8 y^2 z^12+14 x^6 y^4 z^12-16 x^4 y^6 z^12+9 x^2 y^8 z^12-2 y^10 z^12)-32 a^16 (40 x^14 y^10-24 x^12 y^12+16 x^10 y^14+40 x^14 y^8 z^2-299 x^12 y^10 z^2+137 x^10 y^12 z^2-72 x^8 y^14 z^2-80 x^14 y^6 z^4-320 x^12 y^8 z^4+795 x^10 y^10 z^4-351 x^8 y^12 z^4+128 x^6 y^14 z^4-80 x^14 y^4 z^6+166 x^12 y^6 z^6+971 x^10 y^8 z^6-1097 x^8 y^10 z^6+390 x^6 y^12 z^6-112 x^4 y^14 z^6+40 x^14 y^2 z^8-320 x^12 y^4 z^8+971 x^10 y^6 z^8-1155 x^8 y^8 z^8+762 x^6 y^10 z^8-178 x^4 y^12 z^8+48 x^2 y^14 z^8+40 x^14 z^10-299 x^12 y^2 z^10+795 x^10 y^4 z^10-1097 x^8 y^6 z^10+762 x^6 y^8 z^10-274 x^4 y^10 z^10+49 x^2 y^12 z^10-8 y^14 z^10-24 x^12 z^12+137 x^10 y^2 z^12-351 x^8 y^4 z^12+390 x^6 y^6 z^12-178 x^4 y^8 z^12+49 x^2 y^10 z^12-23 y^12 z^12+16 x^10 z^14-72 x^8 y^2 z^14+128 x^6 y^4 z^14-112 x^4 y^6 z^14+48 x^2 y^8 z^14-8 y^10 z^14)+16 a^14 (118 x^14 y^12-2 x^12 y^14+16 x^10 y^16+428 x^14 y^10 z^2-807 x^12 y^12 z^2+172 x^10 y^14 z^2-64 x^8 y^16 z^2-118 x^14 y^8 z^4-2001 x^12 y^10 z^4+1542 x^10 y^12 z^4-398 x^8 y^14 z^4+96 x^6 y^16 z^4-856 x^14 y^6 z^6-1022 x^12 y^8 z^6+3120 x^10 y^10 z^6-1772 x^8 y^12 z^6+280 x^6 y^14 z^6-64 x^4 y^16 z^6-118 x^14 y^4 z^8-1022 x^12 y^6 z^8+3533 x^10 y^8 z^8-2347 x^8 y^10 z^8+817 x^6 y^12 z^8-62 x^4 y^14 z^8+16 x^2 y^16 z^8+428 x^14 y^2 z^10-2001 x^12 y^4 z^10+3120 x^10 y^6 z^10-2347 x^8 y^8 z^10+1040 x^6 y^10 z^10-365 x^4 y^12 z^10+28 x^2 y^14 z^10+118 x^14 z^12-807 x^12 y^2 z^12+1542 x^10 y^4 z^12-1772 x^8 y^6 z^12+817 x^6 y^8 z^12-365 x^4 y^10 z^12-87 x^2 y^12 z^12-18 y^14 z^12-2 x^12 z^14+172 x^10 y^2 z^14-398 x^8 y^4 z^14+280 x^6 y^6 z^14-62 x^4 y^8 z^14+28 x^2 y^10 z^14-18 y^12 z^14+16 x^10 z^16-64 x^8 y^2 z^16+96 x^6 y^4 z^16-64 x^4 y^6 z^16+16 x^2 y^8 z^16)-a^12 (720 x^14 y^14+288 x^12 y^16+10608 x^14 y^12 z^2-3217 x^12 y^14 z^2+288 x^10 y^16 z^2+11216 x^14 y^10 z^4-35778 x^12 y^12 z^4+8666 x^10 y^14 z^4-576 x^8 y^16 z^4-22544 x^14 y^8 z^6-43775 x^12 y^10 z^6+18872 x^10 y^12 z^6-5967 x^8 y^14 z^6-576 x^6 y^16 z^6-22544 x^14 y^6 z^8-37340 x^12 y^8 z^8+34894 x^10 y^10 z^8-20854 x^8 y^12 z^8+2140 x^6 y^14 z^8+288 x^4 y^16 z^8+11216 x^14 y^4 z^10-43775 x^12 y^6 z^10+34894 x^10 y^8 z^10-19615 x^8 y^10 z^10-3050 x^6 y^12 z^10-143 x^4 y^14 z^10+288 x^2 y^16 z^10+10608 x^14 y^2 z^12-35778 x^12 y^4 z^12+18872 x^10 y^6 z^12-20854 x^8 y^8 z^12-3050 x^6 y^10 z^12-6480 x^4 y^12 z^12-966 x^2 y^14 z^12+720 x^14 z^14-3217 x^12 y^2 z^14+8666 x^10 y^4 z^14-5967 x^8 y^6 z^14+2140 x^6 y^8 z^14-143 x^4 y^10 z^14-966 x^2 y^12 z^14-81 y^14 z^14+288 x^12 z^16+288 x^10 y^2 z^16-576 x^8 y^4 z^16-576 x^6 y^6 z^16+288 x^4 y^8 z^16+288 x^2 y^10 z^16)+a^10 x^2 (81 x^12 y^16+3972 x^12 y^14 z^2+1254 x^10 y^16 z^2+22468 x^12 y^12 z^4-8012 x^10 y^14 z^4+1215 x^8 y^16 z^4-3972 x^12 y^10 z^6-34520 x^10 y^12 z^6+1672 x^8 y^14 z^6+84 x^6 y^16 z^6-45098 x^12 y^8 z^8-35778 x^10 y^10 z^8-11013 x^8 y^12 z^8-2152 x^6 y^14 z^8+1215 x^4 y^16 z^8-3972 x^12 y^6 z^10-35778 x^10 y^8 z^10-4764 x^8 y^10 z^10-10400 x^6 y^12 z^10-860 x^4 y^14 z^10+1254 x^2 y^16 z^10+22468 x^12 y^4 z^12-34520 x^10 y^6 z^12-11013 x^8 y^8 z^12-10400 x^6 y^10 z^12-12772 x^4 y^12 z^12-1868 x^2 y^14 z^12+81 y^16 z^12+3972 x^12 y^2 z^14-8012 x^10 y^4 z^14+1672 x^8 y^6 z^14-2152 x^6 y^8 z^14-860 x^4 y^10 z^14-1868 x^2 y^12 z^14-432 y^14 z^14+81 x^12 z^16+1254 x^10 y^2 z^16+1215 x^8 y^4 z^16+84 x^6 y^6 z^16+1215 x^4 y^8 z^16+1254 x^2 y^10 z^16+81 y^12 z^16)-a^8 x^4 y^2 z^2 (432 x^10 y^14+8336 x^10 y^12 z^2+2672 x^8 y^14 z^2+19760 x^10 y^10 z^4-4049 x^8 y^12 z^4+3808 x^6 y^14 z^4-28528 x^10 y^8 z^6-11252 x^8 y^10 z^6-3748 x^6 y^12 z^6+3808 x^4 y^14 z^6-28528 x^10 y^6 z^8-4966 x^8 y^8 z^8-17532 x^6 y^10 z^8+442 x^4 y^12 z^8+2672 x^2 y^14 z^8+19760 x^10 y^4 z^10-11252 x^8 y^6 z^10-17532 x^6 y^8 z^10-9980 x^4 y^10 z^10-2740 x^2 y^12 z^10+432 y^14 z^10+8336 x^10 y^2 z^12-4049 x^8 y^4 z^12-3748 x^6 y^6 z^12+442 x^4 y^8 z^12-2740 x^2 y^10 z^12-945 y^12 z^12+432 x^10 z^14+2672 x^8 y^2 z^14+3808 x^6 y^4 z^14+3808 x^4 y^6 z^14+2672 x^2 y^8 z^14+432 y^10 z^14)+16 a^6 x^6 y^4 z^4 (54 x^8 y^12+492 x^8 y^10 z^2+216 x^6 y^12 z^2+202 x^8 y^8 z^4+121 x^6 y^10 z^4+324 x^4 y^12 z^4-1496 x^8 y^6 z^6+235 x^6 y^8 z^6+3 x^4 y^10 z^6+216 x^2 y^12 z^6+202 x^8 y^4 z^8+235 x^6 y^6 z^8-368 x^4 y^8 z^8-149 x^2 y^10 z^8+54 y^12 z^8+492 x^8 y^2 z^10+121 x^6 y^4 z^10+3 x^4 y^6 z^10-149 x^2 y^8 z^10-75 y^10 z^10+54 x^8 z^12+216 x^6 y^2 z^12+324 x^4 y^4 z^12+216 x^2 y^6 z^12+54 y^8 z^12)-32 a^4 x^8 y^6 z^6 (24 x^6 y^10+88 x^6 y^8 z^2+72 x^4 y^10 z^2-112 x^6 y^6 z^4+29 x^4 y^8 z^4+72 x^2 y^10 z^4-112 x^6 y^4 z^6+10 x^4 y^6 z^6-46 x^2 y^8 z^6+24 y^10 z^6+88 x^6 y^2 z^8+29 x^4 y^4 z^8-46 x^2 y^6 z^8-35 y^8 z^8+24 x^6 z^10+72 x^4 y^2 z^10+72 x^2 y^4 z^10+24 y^6 z^10)+256 a^2 x^10 y^8 z^8 (x^2 y^2+x^2 z^2+y^2 z^2) (x^2 y^6-x^2 y^4 z^2+y^6 z^2-x^2 y^2 z^4-3 y^4 z^4+x^2 z^6+y^2 z^6)+256 x^12 y^14 z^14 = 0
是不是被你搞复杂了？明天我到电脑上看看 数值解是可以的。画一个任意三角形，作出三条角平分线，然后测出 a = 7629, b = 5143, c = 7157, ta = 4759, tb = 6928, tc = 5188。
用下面这程序进行数值解：
Clear["Global`*"];
NSolve[{5188 == Sqrt/(a + b),
6928 == Sqrt/(a + c),
4759 == Sqrt/(c + b), a > 0, b > 0,
c > 0}, {a, b, c}]
结果是：{{a->7629.14,b->5142.95,c->7157.15}} ，完全正确。
结论是： 已知三角形的三条角平分线长 ta, tb, tc 的长度，这个三角形的三边 a, b, c 就唯一地确定了。
至于能不能求出a=f (ta, tb, tc) 的解析式呢？ 还没在哪本数学手册中见过。 chyanog 发表于 2018-7-24 21:18
因为方程组的解确实比较复杂，你可以先带入具体的数值再解（比如ta->3, tb->4, tc->5），这样解起来就快了 ...
Clear["Global`*"];
(*下面三个无理数,不可能被加减乘除消化掉,所以先得出一个伪符号解*)
ta=Pi;
tb=Khinchin;
tc=E;
out=FullSimplify/(a + b),
tb == Sqrt/(a + c),
ta == Sqrt/(c + b), a > 0, b > 0, c > 0
},
{a, b, c},Reals]]
(*回代得到符号解*)
out/.{Pi->ta,Khinchin->tb,E->tc}
为什么我这个得不到结果?是电脑性能问题吗? Clear["Global`*"];
(*下面三个无理数,不可能被加减乘除消化掉,所以先得出一个伪符号解*)
rule={ta,tb,tc}->{\,Khinchin,E}
out=Solve[{tc == Sqrt/(a + b),
tb == Sqrt/(a + c),
ta == Sqrt/(c + b)
}/.Thread,
{a, b, c}]
(*回代得到符号解*)
out/.Thread]
这个代码好像搞不动 对于这个问题,我只能说数值解万岁,理论上能求出来的,但是软件就是算不出来,没办法 已知角平分线的求三边很难,但是中线还算简单
Clear["Global`*"];(*Clear all variables*)
zx:=(a^2+b^2-c^2/2)/2
FullSimplify@Solve[{
ta^2==zx,
tb^2==zx,
tc^2==zx
},{a,b,c}]
\[\left.\left\{a\to \frac{2}{3} \sqrt{2 \left(\text{tb}^2+\text{tc}^2\right)-\text{ta}^2},b\to \frac{2}{3} \sqrt{2 \text{ta}^2-\text{tb}^2+2 \text{tc}^2},c\to \frac{2}{3} \sqrt{2 \left(\text{ta}^2+\text{tb}^2\right)-\text{tc}^2}\right\}\right\}\] Clear["Global`*"];(*Clear all variables*)
(*高的长度*)
gx:=Sqrt[-(a-b-c)*(a+b-c)*(a-b+c)*(a+b+c)]/(2*c)
FullSimplify@Solve[{
ga==gx,
gb==gx,
gc==gx
},{a,b,c}]
高线长度求三边长
\[\left\{a\to \frac{2 \text{ga} \text{gb}^2 \text{gc}^2}{\sqrt{\text{ga}^4 \left(-\left(\text{gb}^2-\text{gc}^2\right)^2\right)+2 \text{ga}^2 \text{gb}^2 \text{gc}^2 \left(\text{gb}^2+\text{gc}^2\right)-\text{gb}^4 \text{gc}^4}},b\to \frac{2 \text{ga}^2 \text{gb} \text{gc}^2}{\sqrt{\text{ga}^4 \left(-\left(\text{gb}^2-\text{gc}^2\right)^2\right)+2 \text{ga}^2 \text{gb}^2 \text{gc}^2 \left(\text{gb}^2+\text{gc}^2\right)-\text{gb}^4 \text{gc}^4}},c\to \frac{2 \text{ga}^2 \text{gb}^2 \text{gc}}{\sqrt{\text{ga}^4 \left(-\left(\text{gb}^2-\text{gc}^2\right)^2\right)+2 \text{ga}^2 \text{gb}^2 \text{gc}^2 \left(\text{gb}^2+\text{gc}^2\right)-\text{gb}^4 \text{gc}^4}}\right\}\]
根号下是
\[-(-\text{ga} \text{gb}-\text{ga} \text{gc}+\text{gb} \text{gc}) (\text{ga} \text{gb}-\text{ga} \text{gc}+\text{gb} \text{gc}) (-\text{ga} \text{gb}+\text{ga} \text{gc}+\text{gb} \text{gc}) (\text{ga} \text{gb}+\text{ga} \text{gc}+\text{gb} \text{gc})\] 本帖最后由 TSC999 于 2018-7-27 08:14 编辑
设 \(W=t_a^2+t_b^2+t_c^2， t=t_a^2t_b^2+t_a^2t_c^2+t_b^2t_c^2，F=t_a^2t_b^2t_c^2\)，未知量是三角形面积的平方，记为\(S2\)，那么理论上可由下面这个关于\(S2\)的 10 次方程求出\(S2\)的解析式。
下面是验证程序。只要程序运行结果等于零，就表明上述结论正确。
Clear["Global`*"];
f = -16777216 F^4 (4 F W - t^2) S2^10 -
2097152 F^2 (-t^6 + 6 F W t^4 + 2 F^2 t^3 - 8 F^2 W^2 t^2 -
8 F^3 W t + 10 F^4) S2^9 +
65536 (t^10 - 8 F W t^8 + 12 F^2 t^7 + 16 F^2 W^2 t^6 -
56 F^3 W t^5 - 120 F^4 t^4 + 64 F^4 W^2 t^3 + 428 F^5 W t^2 +
90 F^6 t - 128 F^5 W^3 t - 112 F^6 W^2) S2^8 -
16384 F (2 W t^9 + 5 F t^8 - 16 F W^2 t^7 + 5 F^2 W t^6 +
58 F^3 t^5 + 32 F^2 W^3 t^5 - 104 F^3 W^2 t^4 - 134 F^4 W t^3 -
392 F^5 t^2 + 32 F^4 W^3 t^2 + 528 F^5 W^2 t - 64 F^5 W^4 +
167 F^6 W) S2^7 -
256 F^2 (-14 t^9 - 16 W^2 t^8 - 24 F W t^7 + 128 F W^3 t^6 -
432 F^2 t^6 + 64 F^2 W^2 t^5 - 256 F^2 W^4 t^4 - 28 F^3 W t^4 -
500 F^4 t^3 + 1024 F^3 W^3 t^3 - 1824 F^4 W^2 t^2 +
10832 F^5 W t + 519 F^6 - 3072 F^5 W^3) S2^6 -
32 F^3 (36 W t^8 + 94 F t^7 - 144 F W^2 t^6 + 744 F^2 W t^5 +
2243 F^3 t^4 - 1536 F^3 W^2 t^3 - 3648 F^4 W t^2 + 9328 F^5 t +
2048 F^4 W^3 t - 7680 F^5 W^2) S2^5 +
F^4 (81 t^8 + 768 F W t^6 + 1568 F^2 t^5 - 3072 F^2 W^2 t^4 +
24064 F^3 W t^3 - 21184 F^4 t^2 - 24576 F^4 W^2 t +
37888 F^5 W) S2^4 +
4 F^6 (-27 t^6 - 32 F W t^4 - 352 F^2 t^3 + 128 F^2 W^2 t^2 -
320 F^3 W t + 424 F^4) S2^3 + 2 F^8 t (27 t^3 + 80 F^2) S2^2 -
12 F^10 t^2 S2^1 + F^12;
f /. {W -> ta2 + tb2 + tc2, t -> ta2 tb2 + tb2 tc2 + tc2 ta2,
F -> ta2 tb2 tc2};
% /. {ta2 -> b c ((b + c)^2 - a^2)/(b + c)^2,
tb2 -> a c ((a + c)^2 - b^2)/(a + c)^2,
tc2 -> b a ((b + a)^2 - c^2)/(b + a)^2,
S2 -> (a + b + c) (-a + b + c) (a - b + c) (a + b - c)/16};
Factor[%]
上面 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。
如果已知三条角平分线的具体数值长度，能够利用上式算出面积平方的数值解。不过大于零的解有 7 个，只有一个是正确的。如果只需要数值解，还不如由平分线先算出三条边长，再求面积。
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