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数学中国»论坛 › 数学中国论坛 › 基础数学 › △ABC中，垂心H，以BC为直径的⊙O上一点满足DH∥BC，DH ...
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△ABC中，垂心H，以BC为直径的⊙O上一点满足DH∥BC，DH的中垂线交⊙O于E、F。∠EHF...

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发表于 2023-4-24 14:08 | 显示全部楼层 |阅读模式
△ABC中，垂心H，以BC为直径的⊙O上一点满足DH∥BC，DH的中垂线交⊙O于E、F。∠EHF的内外角平分线交AC、AB于J、K。求证：JK平分AH。

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发表于 2023-4-25 09:16 | 显示全部楼层
本帖最后由 天山草 于 2023-4-25 14:46 编辑


程序运行结果：

程序代码：
  1. Clear["Global`*"];
  2. u = -((2 (y^2 (z^2 - I z - 1) + z^2 + I z - 1))/((y^2 - 1) (z^2 - 1))); v = -(((u^2 - 1) (z^2 - 1))/( z^2 + 1));(*为使D、E、F点的坐标有理化选择的最终变量y、z*)
  3. \!\(\*OverscriptBox[\(o\), \(_\)]\) = o = 0; \!\(\*OverscriptBox[\(b\), \(_\)]\) = b = -1;
  4. \!\(\*OverscriptBox[\(c\), \(_\)]\) = c = 1; a = u + I v;  \!\(\*OverscriptBox[\(a\), \(_\)]\) = u - I v;
  5. Print["A = ", Simplify[a]]; Print["B = ", b, "，   C = ", c];
  6. kF[a_, b_] := (a - b)/(\!\(\*OverscriptBox[\(a\), \(_\)]\) - \!\(\*OverscriptBox[\(b\), \(_\)]\));(*定义复斜率*)
  7. (*三角形垂心坐标公式：*)
  8. hx [a_, b_, c_] := ((b - c) (b + c - a) \!\(\*OverscriptBox[\(a\), \(_\)]\) + (c - a) (c + a - b) \!\(\*OverscriptBox[\(b\), \(_\)]\) + (a - b) (a + b - c) \!\(\*OverscriptBox[\(c\), \(_\)]\))/((b - c) \!\(\*OverscriptBox[\(a\), \(_\)]\) + (c - a) \!\(\*OverscriptBox[\(b\), \(_\)]\) + (a - b) \!\(\*OverscriptBox[\(c\), \(_\)]\));   
  9. \!\(\*OverscriptBox[\(hx\), \(_\)]\) [a_, b_, c_] := ((\!\(\*OverscriptBox[\(b\), \(_\)]\) - \!\(\*OverscriptBox[\(c\), \(_\)]\)) (\!\(\*OverscriptBox[\(b\), \(_\)]\) + \!\(\*OverscriptBox[\(c\), \(_\)]\) - \!\(\*OverscriptBox[\(a\), \(_\)]\)) a + (\!\(\*OverscriptBox[\(c\), \(_\)]\) - \!\(\*OverscriptBox[\(a\), \(_\)]\)) (\!\(\*OverscriptBox[\(c\), \(_\)]\) + \!\(\*OverscriptBox[\(a\), \(_\)]\) -
  10. \!\(\*OverscriptBox[\(b\), \(_\)]\)) b + (\!\(\*OverscriptBox[\(a\), \(_\)]\) - \!\(\*OverscriptBox[\(b\), \(_\)]\)) (\!\(\*OverscriptBox[\(a\), \(_\)]\) + \!\(\*OverscriptBox[\(b\), \(_\)]\) - \!\(\*OverscriptBox[\(c\), \(_\)]\)) c)/((\!\(\*OverscriptBox[\(b\), \(_\)]\) -
  11. \!\(\*OverscriptBox[\(c\), \(_\)]\)) a + (\!\(\*OverscriptBox[\(c\), \(_\)]\) - \!\(\*OverscriptBox[\(a\), \(_\)]\)) b + (\!\(\*OverscriptBox[\(a\), \(_\)]\) - \!\(\*OverscriptBox[\(b\), \(_\)]\)) c);   
  12. h = Simplify@hx[a, b, c]; \!\(\*OverscriptBox[\(h\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(hx\), \(_\)]\)[a, b, c];
  13. W1 = {d, \!\(\*OverscriptBox[\(d\), \(_\)]\)} /. Simplify@Solve[{(d - o) (\!\(\*OverscriptBox[\(d\), \(_\)]\) - \!\(\*OverscriptBox[\(o\), \(_\)]\)) == 1, kF[d, h] == 1}, {d, \!\(\*OverscriptBox[\(d\), \(_\)]\)}] // Flatten;
  14. d = Part[W1, 1]; \!\(\*OverscriptBox[\(d\), \(_\)]\) = Part[W1, 2]; m = Simplify[(d + h)/2];
  15. \!\(\*OverscriptBox[\(m\), \(_\)]\) = Simplify[(\!\(\*OverscriptBox[\(d\), \(_\)]\) + \!\(\*OverscriptBox[\(h\), \(_\)]\))/2];
  16. W2 = {e, \!\(\*OverscriptBox[\(e\), \(_\)]\)} /. Simplify@Solve[{(e - o) (\!\(\*OverscriptBox[\(e\), \(_\)]\) - \!\(\*OverscriptBox[\(o\), \(_\)]\)) == 1, kF[e, m] == -1}, {e, \!\(\*OverscriptBox[\(e\), \(_\)]\)}] // Flatten;
  17. f = Part[W2, 1]; \!\(\*OverscriptBox[\(f\), \(_\)]\) = Part[W2, 2]; e = Part[W2, 3]; \!\(\*OverscriptBox[\(e\), \(_\)]\) = Part[W2, 4];
  18. Print["H = ", h, "，   D = ", d, "，  M = ", m, "，   E = ", e, "，   F = ", f];
  19. kFH = Simplify@kF[f, h]; kEH = Simplify@kF[e, h];
  20. kKH = Simplify@Sqrt[kFH kEH]; kKH = -((z + 1)/(z - 1)) I;(*KH的复斜率*)
  21. (*过A1点、复斜率等于k1的直线，与过A2点、复斜率等于k2的直线的交点公式：*)
  22. Jd[k1_, a1_, k2_, a2_] := -((k2 (a1 - k1 \!\(\*OverscriptBox[\(a1\), \(_\)]\)) - k1 (a2 - k2 \!\(\*OverscriptBox[\(a2\), \(_\)]\)))/(k1 - k2));
  23. \!\(\*OverscriptBox[\(Jd\), \(_\)]\)[k1_, a1_, k2_, a2_] := -((a1 - k1 \!\(\*OverscriptBox[\(a1\), \(_\)]\) - (a2 - k2 \!\(\*OverscriptBox[\(a2\), \(_\)]\)))/(k1 - k2));
  24. k = Simplify@Jd[kKH, h, kF[a, b], b];
  25. \!\(\*OverscriptBox[\(k\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(Jd\), \(_\)]\)[kKH, h, kF[a, b], b];
  26. j = Simplify@Jd[-kKH, h, kF[a, c], c]; \!\(\*OverscriptBox[\(j\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(Jd\), \(_\)]\)[-kKH, h, kF[a, c], c];
  27. Print["K = ", k, "，   J = ", j];
  28. n = Simplify@Jd[kF[a, h], a, kF[k, j], k] // Factor;
  29. \!\(\*OverscriptBox[\(n\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(Jd\), \(_\)]\)[kF[a, h], a, kF[k, j], k] // Factor;
  30. Print["N = ", n];AN = Simplify[Sqrt[(a - n) (\!\(\*OverscriptBox[\(a\), \(_\)]\) - \!\(\*OverscriptBox[\(n\), \(_\)]\))]];
  31. HN = Simplify[Sqrt[(h - n) (\!\(\*OverscriptBox[\(h\), \(_\)]\) - \!\(\*OverscriptBox[\(n\), \(_\)]\))]];
  32. Print["\!\(\*OverscriptBox[\(AN\), \(_\)]\) = ", AN, "，   \!\(\*OverscriptBox[\(HN\), \(_\)]\) = ", HN];
  33. Print["测试 \!\(\*OverscriptBox[\(AN\), \(_\)]\) = \!\(\*OverscriptBox[\(HN\), \(_\)]\) 是否成立："];
  34. Simplify[AN == HN]
复制代码

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点评

denglongshan
y,z,u和v如何设定需要详细一些，否则不懂  发表于 2023-4-26 19:44
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发表于 2023-4-25 09:29 | 显示全部楼层
本帖最后由 天山草 于 2023-4-25 09:30 编辑

此题用复平面解析几何法做，关键技术是解决圆上点的坐标表达式如何有理化。本程序是采用“欧拉法”有理化的，这个方法是由本论坛 creasson 大侠教给我的，他对有理化有系统的理论研究。期待他的成果不久成书出版。

点评

denglongshan
构图有理化顺序 BC->E->F->->D->H->A->J->K  发表于 2023-4-25 22:29
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发表于 2023-4-26 15:58 | 显示全部楼层
按照楼上  denglongshan 的构图方法，就不存在圆上点的坐标需要有理化的问题。


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发表于 2023-4-27 20:43 | 显示全部楼层
本帖最后由 denglongshan 于 2023-4-27 20:45 编辑

  2. \!\(\*OverscriptBox["o", "_"]\) = o = 0;
  3. \!\(\*OverscriptBox["b", "_"]\) = b = -1;
  4. \!\(\*OverscriptBox["c", "_"]\) = c = 1;
  5. \!\(\*OverscriptBox["d", "_"]\) = 1/d; f =
  6. \!\(\*OverscriptBox["e", "_"]\) = 1/e;
  7. \!\(\*OverscriptBox["f", "_"]\) = 1/f;

  9. KAB[a_, b_] := (a - b)/(
  10. \!\(\*OverscriptBox["a", "_"]\) -
  11. \!\(\*OverscriptBox["b", "_"]\));
  12. \!\(\*OverscriptBox["KAB", "_"]\)[a_, b_] := 1/KAB[a, b];(*复斜率定义*)


  15. \!\(\*OverscriptBox["Jd", "_"]\)[k1_, a1_, k2_, a2_] := -((a1 - k1
  16. \!\(\*OverscriptBox["a1", "_"]\) - (a2 - k2
  17. \!\(\*OverscriptBox["a2", "_"]\)))/(
  18.    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*)
  19. Jd[k1_, a1_, k2_, a2_] := -((k2 (a1 - k1
  20. \!\(\*OverscriptBox["a1", "_"]\)) - k1 (a2 - k2
  21. \!\(\*OverscriptBox["a2", "_"]\)))/(k1 - k2));
  22. FourPoint[a_, b_, c_, d_] := ((
  23. \!\(\*OverscriptBox["c", "_"]\) d - c
  24. \!\(\*OverscriptBox["d", "_"]\)) (a - b) - (
  25. \!\(\*OverscriptBox["a", "_"]\) b - a
  26. \!\(\*OverscriptBox["b", "_"]\)) (c - d))/((a - b) (
  27. \!\(\*OverscriptBox["c", "_"]\) -
  28. \!\(\*OverscriptBox["d", "_"]\)) - (
  29. \!\(\*OverscriptBox["a", "_"]\) -
  30. \!\(\*OverscriptBox["b", "_"]\)) (c - d));(*过两点A和B、C和D的交点*)

  32. \!\(\*OverscriptBox["FourPoint", "_"]\)[a_, b_, c_, d_] := -(((c
  33. \!\(\*OverscriptBox["d", "_"]\) -
  34. \!\(\*OverscriptBox["c", "_"]\) d) (
  35. \!\(\*OverscriptBox["a", "_"]\) -
  36. \!\(\*OverscriptBox["b", "_"]\)) - ( a
  37. \!\(\*OverscriptBox["b", "_"]\) -
  38. \!\(\*OverscriptBox["a", "_"]\) b) (
  39. \!\(\*OverscriptBox["c", "_"]\) -
  40. \!\(\*OverscriptBox["d", "_"]\)))/((a - b) (
  41. \!\(\*OverscriptBox["c", "_"]\) -
  42. \!\(\*OverscriptBox["d", "_"]\)) - (
  43. \!\(\*OverscriptBox["a", "_"]\) -
  44. \!\(\*OverscriptBox["b", "_"]\)) (c - d)));
  45. Duichengdian[a_, b_, p_] := (
  46. \!\(\*OverscriptBox["a", "_"]\) b - a
  47. \!\(\*OverscriptBox["b", "_"]\) +
  48. \!\(\*OverscriptBox["p", "_"]\) (a - b))/(
  49. \!\(\*OverscriptBox["a", "_"]\) -
  50. \!\(\*OverscriptBox["b", "_"]\));
  51. \!\(\*OverscriptBox["Duichengdian", "_"]\)[a_, b_, p_] := (a
  52. \!\(\*OverscriptBox["b", "_"]\) -
  53. \!\(\*OverscriptBox["a", "_"]\) b + p (
  54. \!\(\*OverscriptBox["a", "_"]\) -
  55. \!\(\*OverscriptBox["b", "_"]\)))/(a - b);(*P关于AB的对称点*)
  56. Cpoint[o_, a_, p_] := -KAB[p, a] (
  57. \!\(\*OverscriptBox["a", "_"]\) -
  58. \!\(\*OverscriptBox["o", "_"]\)) + o;

  60. \!\(\*OverscriptBox["Cpoint", "_"]\)[o_, a_, p_] := -
  61. \!\(\*OverscriptBox["KAB", "_"]\)[p, a] (a - o) +
  62. \!\(\*OverscriptBox["o", "_"]\);(*P不在圆上，连接圆O上一点A与圆的另外一个交点*)
  63. h = Duichengdian[e, f, d];
  64. \!\(\*OverscriptBox["h", "_"]\) =
  65. \!\(\*OverscriptBox["Duichengdian", "_"]\)[e, f, d];
  66. n = Cpoint[o, c, h];
  67. \!\(\*OverscriptBox["n", "_"]\) =
  68. \!\(\*OverscriptBox["Cpoint", "_"]\)[o, c, h]; p = Cpoint[o, b, h];
  69. \!\(\*OverscriptBox["p", "_"]\) =
  70. \!\(\*OverscriptBox["Cpoint", "_"]\)[o, b, h];
  71. a = FourPoint[b, n, c, p];
  72. \!\(\*OverscriptBox["a", "_"]\) =
  73. \!\(\*OverscriptBox["FourPoint", "_"]\)[b, n, c, p];
  74. k = Jd[n, b,
  75. \!\(\*OverscriptBox["d", "_"]\), h];
  76. \!\(\*OverscriptBox["k", "_"]\) =
  77. \!\(\*OverscriptBox["Jd", "_"]\)[n, b,
  78. \!\(\*OverscriptBox["d", "_"]\), h]; j = Jd[-p, c, -
  79. \!\(\*OverscriptBox["d", "_"]\), h];
  80. \!\(\*OverscriptBox["j", "_"]\) =
  81. \!\(\*OverscriptBox["Jd", "_"]\)[-p, c, -
  82. \!\(\*OverscriptBox["d", "_"]\), h];(*角EHF平分线复斜率由对称根据图形条件人工推算*)
  83. k1 = Jd[n, b, -
  84. \!\(\*OverscriptBox["d", "_"]\), h];
  85. \!\(\*OverscriptBox["k1", "_"]\) =
  86. \!\(\*OverscriptBox["Jd", "_"]\)[n, b, -
  87. \!\(\*OverscriptBox["d", "_"]\), h]; j1 = Jd[-p, c,
  88. \!\(\*OverscriptBox["d", "_"]\), h];
  89. \!\(\*OverscriptBox["j1", "_"]\) =
  90. \!\(\*OverscriptBox["Jd", "_"]\)[-p, c,
  91. \!\(\*OverscriptBox["d", "_"]\), h];(*角E1HF1平分线复斜率由对称根据图形条件人工推算*)
  92. m = FourPoint[a, h, k, j];
  93. \!\(\*OverscriptBox["m", "_"]\) =
  94. \!\(\*OverscriptBox["FourPoint", "_"]\)[a, h, k, j];
  95. m1 = FourPoint[a, h, k1, j1];
  96. \!\(\*OverscriptBox["m1", "_"]\) =
  97. \!\(\*OverscriptBox["FourPoint", "_"]\)[a, h, k1, j1];
  98. Simplify[{f,
  99. \!\(\*OverscriptBox["f", "_"]\)}]
  100. Simplify[{1, h, , n, p}]
  101. Simplify[{10,
  102. \!\(\*OverscriptBox["h", "_"]\), ,
  103. \!\(\*OverscriptBox["n", "_"]\),
  104. \!\(\*OverscriptBox["p", "_"]\)}]
  105. Simplify[{11, a,
  106. \!\(\*OverscriptBox["a", "_"]\)}]
  107. Simplify[{2, k, , j, m}]
  108. Simplify[{20,
  109. \!\(\*OverscriptBox["k", "_"]\), ,
  110. \!\(\*OverscriptBox["j", "_"]\),
  111. \!\(\*OverscriptBox["m", "_"]\)}]
  112. Simplify[{3, a - m, m - h}]
  113. Simplify[4 ，a - m == m - h]
  114. Simplify[{KAB[a, h], KAB[k, j]}]
  115. Simplify[{5, m1,
  116. \!\(\*OverscriptBox["m1", "_"]\)}]
  117. Simplify[{KAB[a, h], KAB[k1, j1]}]
  118. Simplify[{6, a - h, k1 - j1, , (k1 - j1)/(a - h)}]
  119. Simplify[KAB[a, h] == KAB[k1, j1]]

复制代码




锐角条件多余，交换条件，如果∠EHF的外内角平分线交AC、AB于J1、K1，则从计算结果可得J1K1平行AH，且AH：J1K1=OS：OC。

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