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发表于 2021-12-13 12:44:14 | 显示全部楼层 |阅读模式

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百度几何吧转载https://tieba.baidu.com/p/7451652229
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 2021-12-17 19:18:49 | 显示全部楼层
本帖最后由 dlsh 于 2021-12-17 19:20 编辑


贴吧网友qzc的解答:
只要证明这个结论:
S_1S_2S_3是S关于△ABC的塞瓦三角形,BU,CV分别平行S_1S_2,S_1S_3交对边于U,V.UV交BC于T,则AT平行S的三线性极线.
证明:设BU∩CV=W,G为△ABC的重心,作过A,B,C,G,S的二次曲线Γ,设A,B,C关于Γ的极点分别为A',B',C',则A'在S_2S_3上等.下面证明W∈B'C'.
让S在Γ上运动,则BU∩B'C'→C'S_2→BS→S→CV∩B'C'是射影对应,则只要考虑三种情况使它们相等.
当S=G时显然成立.
当S=C时,BU∥CC',故BU=BB',CV∥CB',故CU=CB',此时结论成立
当S=B时同上可知结论成立
设B'C'∩BC=T*,则由W∈B'C'知[B,C;T,T*]=-1,故A,A',T共线,故AT和S的三线性极线平行.
看不懂,图片链接不出来,只好上传。

贴吧原图

贴吧原图
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 2021-12-21 12:53:51 | 显示全部楼层
如果先构造P,再作R点,利用Ceva定理,出现高次方程,先用Ceva定理,求出三边比例 关系,出现Ceva点不同,十分费解。
  1. (*Clear["Global`*"]*)


  2. \!\(\*OverscriptBox["a", "_"]\) = 1/a;
  3. \!\(\*OverscriptBox["b", "_"]\) = 1/b;
  4. \!\(\*OverscriptBox["c", "_"]\) = 1/c;(*
  5. \!\(\*OverscriptBox["p", "_"]\)=-((-a+2 b-c-p)/(a c ))*);

  6. k[a_, b_] := (a - b)/(
  7. \!\(\*OverscriptBox["a", "_"]\) -
  8. \!\(\*OverscriptBox["b", "_"]\));
  9. \!\(\*OverscriptBox["k", "_"]\)[a_, b_] := 1/k[a, b];(*复斜率定义*)

  10. \!\(\*OverscriptBox["Jd", "_"]\)[k1_, a1_, k2_, a2_] := -((a1 - k1
  11. \!\(\*OverscriptBox["a1", "_"]\) - (a2 - k2
  12. \!\(\*OverscriptBox["a2", "_"]\)))/(
  13.    k1 - k2));(*复斜率等于k1,过点A1与复斜率等于k2,过点A2的直线交点*)
  14. Jd[k1_, a1_, k2_, a2_] := -((k2 (a1 - k1
  15. \!\(\*OverscriptBox["a1", "_"]\)) - k1 (a2 - k2
  16. \!\(\*OverscriptBox["a2", "_"]\)))/(k1 - k2));
  17. Chuizu[a_, b_, p_] := (
  18. \!\(\*OverscriptBox["a", "_"]\) b - a
  19. \!\(\*OverscriptBox["b", "_"]\) + p (
  20. \!\(\*OverscriptBox["a", "_"]\) -
  21. \!\(\*OverscriptBox["b", "_"]\)) +
  22. \!\(\*OverscriptBox["p", "_"]\) (a - b))/(2 (
  23. \!\(\*OverscriptBox["a", "_"]\) -
  24. \!\(\*OverscriptBox["b", "_"]\)));(*=(1/2)[p+(
  25. \!\(\*OverscriptBox["a", "_"]\)b-a
  26. \!\(\*OverscriptBox["b", "_"]\)+
  27. \!\(\*OverscriptBox["p", "_"]\)(a-b))/(
  28. \!\(\*OverscriptBox["a", "_"]\)-
  29. \!\(\*OverscriptBox["b", "_"]\))]P到直线AB的垂足*)

  30. \!\(\*OverscriptBox["Chuizu", "_"]\)[a_, b_, p_] := (a
  31. \!\(\*OverscriptBox["b", "_"]\) -
  32. \!\(\*OverscriptBox["a", "_"]\) b +
  33. \!\(\*OverscriptBox["p", "_"]\) (a - b) + p (
  34. \!\(\*OverscriptBox["a", "_"]\) -
  35. \!\(\*OverscriptBox["b", "_"]\)))/(2 (a - b));
  36. FourPoint[a_, b_, c_, d_] := ((
  37. \!\(\*OverscriptBox["c", "_"]\) d - c
  38. \!\(\*OverscriptBox["d", "_"]\)) (a - b) - (
  39. \!\(\*OverscriptBox["a", "_"]\) b - a
  40. \!\(\*OverscriptBox["b", "_"]\)) (c - d))/((a - b) (
  41. \!\(\*OverscriptBox["c", "_"]\) -
  42. \!\(\*OverscriptBox["d", "_"]\)) - (
  43. \!\(\*OverscriptBox["a", "_"]\) -
  44. \!\(\*OverscriptBox["b", "_"]\)) (c - d));(*过两点A和B、C和D的交点*)

  45. \!\(\*OverscriptBox["FourPoint", "_"]\)[a_, b_, c_, d_] := -(((c
  46. \!\(\*OverscriptBox["d", "_"]\) -
  47. \!\(\*OverscriptBox["c", "_"]\) d) (
  48. \!\(\*OverscriptBox["a", "_"]\) -
  49. \!\(\*OverscriptBox["b", "_"]\)) - ( a
  50. \!\(\*OverscriptBox["b", "_"]\) -
  51. \!\(\*OverscriptBox["a", "_"]\) b) (
  52. \!\(\*OverscriptBox["c", "_"]\) -
  53. \!\(\*OverscriptBox["d", "_"]\)))/((a - b) (
  54. \!\(\*OverscriptBox["c", "_"]\) -
  55. \!\(\*OverscriptBox["d", "_"]\)) - (
  56. \!\(\*OverscriptBox["a", "_"]\) -
  57. \!\(\*OverscriptBox["b", "_"]\)) (c - d)));
  58. f = Chuizu[a, b, p];
  59. \!\(\*OverscriptBox["f", "_"]\) =
  60. \!\(\*OverscriptBox["Chuizu", "_"]\)[a , b, p]; e = Chuizu[a , c, p];
  61. \!\(\*OverscriptBox["e", "_"]\) =
  62. \!\(\*OverscriptBox["Chuizu", "_"]\)[a , c, p]; d = Chuizu[b, c, p];
  63. \!\(\*OverscriptBox["d", "_"]\) =
  64. \!\(\*OverscriptBox["Chuizu", "_"]\)[c, b, p];
  65. r1 = FourPoint[f, c, b, e];
  66. \!\(\*OverscriptBox["r1", "_"]\) =
  67. \!\(\*OverscriptBox["FourPoint", "_"]\)[f, c, b, e]; r =
  68. FourPoint[a, d, b, e];
  69. \!\(\*OverscriptBox["r", "_"]\) =
  70. \!\(\*OverscriptBox["FourPoint", "_"]\)[a, d, b, e];
  71. u = Jd[-k[c, p], b, -a c, a];
  72. \!\(\*OverscriptBox["u", "_"]\) =
  73. \!\(\*OverscriptBox["Jd", "_"]\)[-k[c, p], b, -a c, a]; v =
  74. Jd[-k[b, p], c, -a b, a];
  75. \!\(\*OverscriptBox["v", "_"]\) =
  76. \!\(\*OverscriptBox["Jd", "_"]\)[-k[b, p], c, -a b, a];
  77. t = FourPoint[b, c, u, v];
  78. \!\(\*OverscriptBox["t", "_"]\) =
  79. \!\(\*OverscriptBox["FourPoint", "_"]\)[b, c, u, v];

  80. Simplify[{r, r1, , r1 - r}](*验证Ceva点*)
  81. Factor[{1, r, r1, , r1 - r}]
  82. Simplify[{2, u,
  83. \!\(\*OverscriptBox["u", "_"]\), , v, v, , t, t, , k[a, t], k[p, r], ,
  84.    k[a, t] - k[p, r]}](*k[p,r],k[p,r]+k[a,t]*)
  85. Factor[{2, u,
  86. \!\(\*OverscriptBox["u", "_"]\), , v, v, , t, t, , k[a, t],
  87.   k[p, r]}](*k[p,r],k[p,r]+k[a,t]*)
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表达式太长,就不上传了,感觉不对。
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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