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[转载] 欧拉线有关的共线

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发表于 2022-6-1 20:59:58 | 显示全部楼层 |阅读模式

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三角形ABC,BC处关于(O)的切线交于D,AE垂直于Euler line且交(O)于另一点E,DE交(O)于另一点F.
设D关于BC的对称点位D'.证明:AD'F共线.
转自纯几何吧
微信图片_20220601205434.jpg
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
发表于 2022-6-3 11:38:16 | 显示全部楼层
非常适合用复平面解析几何方法做此题。

证明 A D' F 共线.png

点评

抱歉,一直没有回复  发表于 2023-5-10 20:59
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 2023-5-10 21:01:10 | 显示全部楼层
本帖最后由 dlsh 于 2023-5-10 21:03 编辑

  1. \!\(\*OverscriptBox["o", "_"]\) = o = 0;
  2. \!\(\*OverscriptBox["a", "_"]\) = 1/a;
  3. \!\(\*OverscriptBox["b", "_"]\) = 1/b;
  4. \!\(\*OverscriptBox["c", "_"]\) = 1/c; h = a + b + c;
  5. \!\(\*OverscriptBox["h", "_"]\) =
  6. \!\(\*OverscriptBox["a", "_"]\) +
  7. \!\(\*OverscriptBox["b", "_"]\) +
  8. \!\(\*OverscriptBox["c", "_"]\);(*外心在原点,并且是单位圆,H是垂心*)
  9. d = (2 b c)/(b + c);
  10. \!\(\*OverscriptBox["d", "_"]\) = 2/(b + c);(*D是过B和C的交点*)
  11. KAB[a_, b_] := (a - b)/(
  12. \!\(\*OverscriptBox["a", "_"]\) -
  13. \!\(\*OverscriptBox["b", "_"]\));
  14. \!\(\*OverscriptBox["KAB", "_"]\)[a_, b_] := 1/KAB[a, b];(*复斜率定义*)
  15. k0 = KAB[o, h];(*欧拉直线复斜率*)
  16. e = k0/a;
  17. \!\(\*OverscriptBox["e", "_"]\) = 1/e;
  18. f = -KAB[d, e]/e;
  19. \!\(\*OverscriptBox["f", "_"]\) = 1/f;
  20. Duichengdian[a_, b_, p_] := (
  21. \!\(\*OverscriptBox["a", "_"]\) b - a
  22. \!\(\*OverscriptBox["b", "_"]\) +
  23. \!\(\*OverscriptBox["p", "_"]\) (a - b))/(
  24. \!\(\*OverscriptBox["a", "_"]\) -
  25. \!\(\*OverscriptBox["b", "_"]\));
  26. \!\(\*OverscriptBox["Duichengdian", "_"]\)[a_, b_, p_] := (a
  27. \!\(\*OverscriptBox["b", "_"]\) -
  28. \!\(\*OverscriptBox["a", "_"]\) b + p (
  29. \!\(\*OverscriptBox["a", "_"]\) -
  30. \!\(\*OverscriptBox["b", "_"]\)))/(a - b);(*P关于AB的对称点*)
  31. d' = Duichengdian[b, c, d];
  32. \!\(\*OverscriptBox[
  33. RowBox[{"d", "'"}], "_"]\) =
  34. \!\(\*OverscriptBox["Duichengdian", "_"]\)[b, c, d];
  35. Print["欧拉直线的复斜率=", Simplify[k0]];
  36. Print["d=", Simplify[d], "     \!\(\*OverscriptBox["d", "_"]\)=",
  37.   Simplify[
  38. \!\(\*OverscriptBox["d", "_"]\)]];
  39. Print["e=", Simplify[e], "    \!\(\*OverscriptBox["e", "_"]\)=",
  40.   Simplify[
  41. \!\(\*OverscriptBox["e", "_"]\)]];
  42. Print["f=", Simplify[f], "    \!\(\*OverscriptBox["f", "_"]\)=",
  43.   Simplify[
  44. \!\(\*OverscriptBox["f", "_"]\)]];
  45. Print["d'=", Simplify[d'], "   \!\(\*OverscriptBox[
  46. RowBox[{"d", "'"}], "_"]\)=", Simplify[
  47. \!\(\*OverscriptBox[
  48. RowBox[{"d", "'"}], "_"]\)]];
  49. Print["AF的复斜率=", Simplify[KAB[a, f]], "   AD'的复斜率=",
  50. Simplify[KAB[a, d']]]
  51. Print["AF和AD'的复斜率相等,所以这三点共线"]
  52. Print["\!\(\*FractionBox[OverscriptBox[
  53. RowBox[{"AD", "'"}], "\[RightVector]"], OverscriptBox["AF", \
  54. "\[RightVector]"]]\)=", Simplify[(d' - a)/(f - a)],
  55.   "     \!\(\*OverscriptBox[
  56. RowBox[{" ",
  57. RowBox[{"(", FractionBox[OverscriptBox[
  58. RowBox[{"AD", "'"}], "\[RightVector]"], OverscriptBox["AF", \
  59. "\[RightVector]"]], ")"}]}], "_"]\)=", Simplify[(
  60. \!\(\*OverscriptBox[
  61. RowBox[{"d", "'"}], "_"]\) -
  62. \!\(\*OverscriptBox["a", "_"]\))/(
  63. \!\(\*OverscriptBox["f", "_"]\) -
  64. \!\(\*OverscriptBox["a", "_"]\))]]
  65. Print["\!\(\*FractionBox[OverscriptBox[
  66. RowBox[{"AD", "'"}], "\[RightVector]"], OverscriptBox["AF", \
  67. "\[RightVector]"]]\)=\!\(\*OverscriptBox[
  68. RowBox[{"(", FractionBox[OverscriptBox[
  69. RowBox[{"AD", "'"}], "\[RightVector]"], OverscriptBox["AF", \
  70. "\[RightVector]"]], ")"}], \
  71. "_"]\)相等,说明\!\(\*FractionBox[OverscriptBox[
  72. RowBox[{"AD", "'"}], "\[RightVector]"], OverscriptBox["AF", \
  73. "\[RightVector]"]]\)虚部等于0,所以这三点共线"](*验证虚部等于0*)
复制代码
与欧拉直线有关的共线问题.gif

点评

复斜率比向量商的表达式简单得多  发表于 2023-5-10 21:17
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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