几何超难题求解

iseemu2009

如图所示为三角形中心大全网站使用的(6,9,13)三角形，AB=6，BC=13，CA=9，D、E 是 BC边上的两点，满足三角形ABD，ADE，AEC 的内切圆半径相等，求半径的值？

王守恩

1. N[Solve[{6/Sin[2 a] == 9/Sin[2 b] == 13/Sin[2 a + 2 b], Cos[b]/Sin[b] + Cos[B]/Sin[B] == 6/r, Cos[a]/Sin[a] + Cos[A]/Sin[A] == 9/r,
   
2.    Cos[b]/Sin[b] + Sin[b + B]/Cos[b + B] + Cos[b + B]/Sin[b + B] + Cos[a + A]/Sin[a + A] + Sin[a + A]/Cos[a + A] + Cos[a]/Sin[a] == 13/r, 1 > B > b > A > a > 0}, {a, A, b, B, r}], 20]

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{{a -> 0.20822601201426420706, b -> 0.32600003255365010672, x -> 0.26278266809129013173, y -> 0.35768642953119268949, r -> 1.0650522444110092760}}

nyy

如图。你的图在哪里？

wayne

答案是$\frac{2}{13} \sqrt{10 (14+14^{1/3}-2* 14^{2/3})}$, 即$28561 t^6-283920 t^4+1075200 t^2-896000=0$的一个根，$1.06505224441100927598477343501$

王守恩

我的软件就是出不来答案。6, 9, 13应该是可以换的。

1. Solve[{6/Sin[2a]==9/Sin[2b]==13/Sin[2(a + b)], Cot[b] + Cot[B]==6/r, Cot[a] + Cot[A]==9/r, Cot[b] + Cot[a] + 2/Sin[2(b + B)] + 2/Sin[2(a + A)]==13/r, 1>B>b>A>a>0},{a,A,b,B,r}]

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iseemu2009

wayne 发表于 2024-11-23 22:13
答案是$\frac{2}{13} \sqrt{10 (14+14^{1/3}-2* 14^{2/3})}$, 即$28561 t^6-283920 t^4+1075200 t^2-896000 ...

你是答案是正确的。来个更难一点的问题：三角形 ABC 中，AB=19，BC=26，CA=15。D、E、F、G、H 在边BC上，且它们从 B 到 C 按顺序排列。连接 AD、AE、AF、AG、AH，使相邻的 6 个小三角形有相等的内切圆半径，求 AF 的长度。

nyy

1. Clear["Global`*"];(*mathematica11.2,win7(64bit)Clear all variables*)
   
2. (*子函数,利用三边计算角的余弦值,角是c边所对的角*)
   
3. cs[a_,b_,c_]:=((a^2+b^2-c^2)/(2*a*b))
   
4. (*子函数,海伦公式,利用海伦公式计算三角形的面积*)
   
5. heron[a_,b_,c_]:=Module[{p=(a+b+c)/2},Sqrt[p*(p-a)*(p-b)*(p-c)]]
   
6. (*AE=a,AD=b,CE=c,ED=d,DB=e*)
   
7. ans=NSolve[{
   
8.     (heron[9,a,c]/(a+c+9))^2==(heron[a,b,d]/(a+b+d))^2==(heron[b,e,6]/(b+e+6))^2,(*三角形的内切圆半径相等*)
   
9.     c+d+e==13,(*BC的线段长度=13*)
   
10.     cs[a,c,9]+cs[a,d,b]==0,(*∠AEC的余弦值+∠AED的余弦值=0*)
    
11.     cs[b,d,a]+cs[b,e,6]==0,(*余弦值相加等于零*)
    
12.     a>0&&b>0&&c>0&&d>0&&e>0(*限制变量范围*)
    
13. },{a,b,c,d,e},1000](*先求出数值解*)
    
14. aaa=RootApproximant[{a,b,c,d,e}/.ans](*由数值解得到精确解*)
    
15. bbb=ToRadicals[aaa](*转化为根式*)
    

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求解结果

1. {{a -> 4.5016151684142909712844132436327097633024219506621540387271314\
   
2. 3179447165200903881834182753390035778908460659730014706227592687244440\
   
3. 4038318353261741079864607320984923773816296746195329432956879094655905\
   
4. 0848478084772294338545324397269168271963851847401152250131480408335910\
   
5. 1253799481544561758927520501078214660419166377265690564263168380517729\
   
6. 3181546564780297030962031891534475281359053124680827894437169972540329\
   
7. 9852240491721382720723534066860324524980177567643516677673862273649676\
   
8. 1041848545550881332029356824413324033062488310277315417575489442873883\
   
9. 7831453967345046110101962740028374544480691799505463965328106712867754\
   
10. 6256396155833000337896766932246063637497434114445770461277384537302860\
    
11. 4384038761851648002727182585049890592365634867149834057373046033107236\
    
12. 7216890981334224632491781325000721719032034311110983553408352329605249\
    
13. 568381490776597018766138908058424918375631787747448736882524480278,
    
14.   b -> 3.7173128293246431018508969026216709378219817029066716445160095\
    
15. 3627652918010162194914712152727277866836753728471914485292061087960904\
    
16. 8875352036871829541225421696643714238569976861616872258810847292516060\
    
17. 1658789338513973047682815654750201430595130578364466109609617871732067\
    
18. 8456400381707274278133395633642707381606746502247789053644579024343056\
    
19. 1464018261630781134806168728048316118659071674184506198286342163000792\
    
20. 3711618446871303330159266582164818498444554639174219093897397567286753\
    
21. 9971576695606002109801314856846791369649663008169367882321855292760859\
    
22. 8330740055135458092382820279164745785335229231136263452831997395363309\
    
23. 5958200546956161652249259887194186407695757806011441677681830515287442\
    
24. 6956934762798559677359596485503715574079905235889636920678381789410351\
    
25. 4517491889594916410709168973909118078910152211857999727976372090522018\
    
26. 438895824920625654921460604302965393497151265825701034395924251707,
    
27.   c -> 5.5831096501820397872758711160621155408164867339385133875529760\
    
28. 3775421429177757182306018132473565257027879475943124358182285586464543\
    
29. 4631226397678241210002145747790741620921736885076037465941167794279724\
    
30. 6338169167599580709848496949309234313918657226934736884796607010171010\
    
31. 9357022478951690915541598105011679121826510579568632412106265027727348\
    
32. 2665738888846565513560837433264025370558205277901713160461519114973692\
    
33. 8463591589397466776111456272374784542367399169581142133303027257131260\
    
34. 1666708079344916159833139260896345400754143103560727139013586230899823\
    
35. 8447687213259680604911562524888572522257353034228584187496366517534220\
    
36. 6662481689419382105310988051426544843035615084493378542586133779565149\
    
37. 6573726441515079537078165865919409825791049120380403142258190928366257\
    
38. 6387078303303777450383717435581159315870798032301601076532472048894724\
    
39. 705787520885459428808473987872754198572842875808542183823827215176,
    
40.   d -> 3.3986434693884741008785708110478349104152410736070903749148615\
    
41. 4724441737826547309984372602871950952310730035451767624053970263561987\
    
42. 2372854037689616667436471038811907986067386166506647754632804475939327\
    
43. 9821984567119392260404204550915522979264458832492306608928070991949983\
    
44. 2122062765961579083441207758887198208187152791744239879347220543423583\
    
45. 7442622646981235550008740375106689705033252952150727349986920508004662\
    
46. 9942695527683677359111825767013859448321032690033622863031347060905995\
    
47. 7971178016427809962988181859454974874788909642467772652765747983823103\
    
48. 7836426952908214743449617330409057956297004462933202220816473709185928\
    
49. 4625514305132967637805863861891467995807264003215424728914067428733476\
    
50. 8850783995896716076592873098033425079238161735460854259010211722686502\
    
51. 8364062730870335627724653521396949107194822430096263243538581036027334\
    
52. 894437885375875909993605982940324607806748928327573377441934323806,
    
53.   e -> 4.0182468804294861118455580728900495487682721924543962375321624\
    
54. 1500136832995695507709609264654483790661390488605108017763744149973469\
    
55. 2995919564632142122561383213397350393010876948417314779426027729780947\
    
56. 3839846265281027029747298499775242706816883940572956506275321997879005\
    
57. 8520914755086730001017194136101122669986336628687127708546514428849067\
    
58. 9891638464172198936430422191629284924408541769947559489551560377021644\
    
59. 1593712882918855864776717960611356009311568140385235003665625681962744\
    
60. 0362113904227273877178678879648679724456947253971500208220665785277072\
    
61. 3715885833832104651638820144702369521445642502838213591687159773279850\
    
62. 8712004005447650256883148086681987161157120912291196728499798791701373\
    
63. 4575489562588204386328961036047165094970789144158742598731597348947239\
    
64. 5248858965825886921891629043021891576934379537602135679928946915077940\
    
65. 399774593738664661197920029186921193620408195863884438734238461019}}

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精确解是
\[\left(
\begin{array}{ccccc}
\text{Root}\left[169 \text{$\#$1}^3-1680 \text{$\#$1}-7854\&,1\right] & \text{Root}\left[169 \text{$\#$1}^3-1680 \text{$\#$1}-2436\&,1\right] & \text{Root}\left[169 \text{$\#$1}^3-4173 \text{$\#$1}^2+36027 \text{$\#$1}-100477\&,1\right] & \text{Root}\left[\text{$\#$1}^3+42 \text{$\#$1}-182\&,1\right] & \text{Root}\left[169 \text{$\#$1}^3-2418 \text{$\#$1}^2+13212 \text{$\#$1}-25012\&,1\right] \\
\end{array}
\right)\]
六个根
{{Root[-7854 - 1680 #1 + 169 #1^3 &, 1],
  Root[-2436 - 1680 #1 + 169 #1^3 &, 1],
  Root[-100477 + 36027 #1 - 4173 #1^2 + 169 #1^3 &, 1],
  Root[-182 + 42 #1 + #1^3 &, 1],
  Root[-25012 + 13212 #1 - 2418 #1^2 + 169 #1^3 &, 1]}}

对应的根式形式是
\[\left(
\begin{array}{ccccc}
\frac{5 \sqrt[3]{14}}{13}+\frac{8\ 14^{2/3}}{13} & \frac{8 \sqrt[3]{14}}{13}+\frac{5\ 14^{2/3}}{13} & \frac{1}{13} \left(107+5 \sqrt[3]{14}-8\ 14^{2/3}\right) & -\sqrt[3]{14}+14^{2/3} & \frac{1}{13} \left(62+8 \sqrt[3]{14}-5\ 14^{2/3}\right) \\
\end{array}
\right)\]

nyy

nyy 发表于 2025-1-24 14:44
求解结果

1. aaa=RootApproximant[{a,b,c,d,e}/.ans](*由数值解得到精确解*)
   
2. bbb=ToRadicals[aaa](*转化为根式*)
   
3. rule=Thread[{a,b,c,d,e}->bbb[[1]]](*替换规则*)
   
4. Grid[Transpose@{rule},Alignment->Left](*列表显示*)
   
5. ccc=RootReduce[heron[9,a,c]/(a+c+9)/.rule](*找到内切圆的半径的根*)
   
6. ddd=ToRadicals[ccc](*转化为根式*)
   

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六边的长度
\[\begin{array}{l}
a\to \frac{5 \sqrt[3]{14}}{13}+\frac{8\ 14^{2/3}}{13} \\
b\to \frac{8 \sqrt[3]{14}}{13}+\frac{5\ 14^{2/3}}{13} \\
c\to \frac{1}{13} \left(107+5 \sqrt[3]{14}-8\ 14^{2/3}\right) \\
d\to 14^{2/3}-\sqrt[3]{14} \\
e\to \frac{1}{13} \left(62+8 \sqrt[3]{14}-5\ 14^{2/3}\right) \\
\end{array}\]

圆的半径的方程的根(结果错误，应该乘以2)
Root[-14000 + 67200 #1^2 - 70980 #1^4 + 28561 #1^6 &, 2]
代数表达式
\[\frac{1}{13} \sqrt{10 \left(14+\sqrt[3]{14}-2\ 14^{2/3}\right)}\]

半径
Root[-896000 + 1075200 #1^2 - 283920 #1^4 + 28561 #1^6 &, 2]
代数表达式
\[\frac{2}{13} \sqrt{10 \left(14+\sqrt[3]{14}-2\ 14^{2/3}\right)}\]

数值等于
1.0650522444110092760

修正后的代码

1. Clear["Global`*"];(*mathematica11.2,win7(64bit)Clear all variables*)
   
2. (*子函数,利用三边计算角的余弦值,角是c边所对的角*)
   
3. cs[a_,b_,c_]:=((a^2+b^2-c^2)/(2*a*b))
   
4. (*子函数,海伦公式,利用海伦公式计算三角形的面积*)
   
5. heron[a_,b_,c_]:=Module[{p=(a+b+c)/2},Sqrt[p*(p-a)*(p-b)*(p-c)]]
   
6. (*AE=a,AD=b,CE=c,ED=d,DB=e*)
   
7. ans=NSolve[{
   
8.     (heron[9,a,c]/(a+c+9))^2==(heron[a,b,d]/(a+b+d))^2==(heron[b,e,6]/(b+e+6))^2,(*三角形的内切圆半径相等*)
   
9.     c+d+e==13,(*BC的线段长度=13*)
   
10.     cs[a,c,9]+cs[a,d,b]==0,(*∠AEC的余弦值+∠AED的余弦值=0*)
    
11.     cs[b,d,a]+cs[b,e,6]==0,(*余弦值相加等于零*)
    
12.     a>0&&b>0&&c>0&&d>0&&e>0(*限制变量范围*)
    
13. },{a,b,c,d,e},1000](*先求出数值解*)
    
14. aaa=RootApproximant[{a,b,c,d,e}/.ans](*由数值解得到精确解*)
    
15. bbb=ToRadicals[aaa](*转化为根式*)
    
16. rule=Thread[{a,b,c,d,e}->bbb[[1]]](*替换规则*)
    
17. Grid[Transpose@{rule},Alignment->Left](*列表显示*)
    
18. ccc=RootReduce[2*heron[9,a,c]/(a+c+9)/.rule](*找到内切圆的半径的根*)
    
19. ddd=ToRadicals[ccc](*转化为根式*)
    

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nyy

iseemu2009 发表于 2025-1-24 11:35
你是答案是正确的。来个更难一点的问题：三角形 ABC 中，AB=19，BC=26，CA=15。D、E、F、G、H 在边BC上， ...

你有答案吗？

nyy

iseemu2009 发表于 2025-1-24 11:35
你是答案是正确的。来个更难一点的问题：三角形 ABC 中，AB=19，BC=26，CA=15。D、E、F、G、H 在边BC上， ...

1. Clear["Global`*"];(*mathematica11.2,win7(64bit)Clear all variables*)
   
2. (*子函数,利用三边计算角的余弦值,角是c边所对的角*)
   
3. cs[a_,b_,c_]:=((a^2+b^2-c^2)/(2*a*b))
   
4. (*子函数,海伦公式,利用海伦公式计算三角形的面积*)
   
5. heron[a_,b_,c_]:=Module[{p=(a+b+c)/2},Sqrt[p*(p-a)*(p-b)*(p-c)]]
   
6. (*子函数,用来计算内切圆半径的平方*)
   
7. rr[a_,b_,c_]:=((heron[a,b,c]/(a+b+c))^2)
   
8. (*子函数,用来计算平角两侧的余弦值的和,最后同分得到分子*)
   
9. pj[a_,b_,c_,d_,e_]:=(Numerator@Together[cs[a,c,d]+cs[b,c,e]]==0)
   
10. ans=NSolve[{
    
11.     rr[19,a,f]==rr[a,b,g]==rr[b,c,h]==rr[c,d,i]==rr[d,e,j]==rr[e,15,k],
    
12.     pj[f,g,a,19,b],pj[g,h,b,a,c],pj[h,i,c,b,d],pj[i,j,d,c,e],pj[j,k,e,d,15],
    
13.     a+b+c+d+e+f+g+h+i+j+k==26,
    
14.     a>0&&b>0&&c>0&&d>0&&e>0&&f>0&&g>0&&h>0&&i>0&&j>0&&k>0(*限制变量范围*)
    
15. },{a,b,c,d,e,f,g,h,i,j,k},1000](*先求出数值解*)
    
16. aaa=RootApproximant[{a,b,c,d,e,f,g,h,i,j,k}/.ans](*由数值解得到精确解*)
    
17. bbb=ToRadicals[aaa](*转化为根式*)
    
18. rule=Thread[{a,b,c,d,e,f,g,h,i,j,k}->bbb[[1]]](*替换规则*)
    
19. Grid[Transpose@{rule},Alignment->Left](*列表显示*)
    
20. ccc=RootReduce[2*heron[19,a,f]/(19+a+f)/.rule](*找到内切圆的半径的根*)
    
21. ddd=ToRadicals[ccc](*转化为根式*)
    

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你的回答，计算量太大，软件没计算出结果