ΔABC中，BE,CF⊥∠A平分线，D,G为BC,AC中点，ΔDEF外接圆交BC于H，证：HIDG四点共圆

渣k

用三角或复数会不会简便些

denglongshan

AC＞AB多余，用向量商获得红色标记的结果

1. 
   
2. \!\(\*OverscriptBox["b", "_"]\) = b = 0;
   
3. \!\(\*OverscriptBox["c", "_"]\) = c = 1; a = 1/(1 - \[Lambda] v);
   
4. \!\(\*OverscriptBox["a", "_"]\) = v/(v - \[Lambda]);(*
   
5. \!\(\*OverscriptBox["AC", "\[RightVector]"]\)/
   
6. \!\(\*OverscriptBox["AB", "\[RightVector]"]\)=\[Lambda]v*)
   
7. 
   
8. \!\(\*OverscriptBox["d", "_"]\) = d = 1/2;
   
9. \!\(\*OverscriptBox["h", "_"]\) = h; g = (a + c)/2;
   
10. \!\(\*OverscriptBox["g", "_"]\) = (
    
11. \!\(\*OverscriptBox["a", "_"]\) +
    
12. \!\(\*OverscriptBox["c", "_"]\))/2;
    
13. k[a_, b_] := (a - b)/(
    
14. \!\(\*OverscriptBox["a", "_"]\) -
    
15. \!\(\*OverscriptBox["b", "_"]\));
    
16. \!\(\*OverscriptBox["k", "_"]\)[a_, b_] := 1/k[a, b];(*复斜率定义*)
    
17. k[a_, b_, c_] := k[a, b]/k[b, c];(*e^(2iB) 等于复斜率相除*)
    
18. Chuizuk[a_, k_, p_] := 1/2 (p + k
    
19. \!\(\*OverscriptBox["p", "_"]\) + a - k
    
20. \!\(\*OverscriptBox["a", "_"]\));(*P到过点A，复斜率为k直线的垂足*)
    
21. 
    
22. \!\(\*OverscriptBox["Chuizuk", "_"]\)[a_, k_, p_] := 1/2 (
    
23. \!\(\*OverscriptBox["p", "_"]\) + p/k +
    
24. \!\(\*OverscriptBox["a", "_"]\) - a/k);
    
25. Waixin[a_, b_, c_] := (a
    
26. \!\(\*OverscriptBox["a", "_"]\) (b - c) + b
    
27. \!\(\*OverscriptBox["b", "_"]\) (c - a) + c
    
28. \!\(\*OverscriptBox["c", "_"]\) (a - b) )/(
    
29. \!\(\*OverscriptBox["a", "_"]\) (b - c) +
    
30. \!\(\*OverscriptBox["b", "_"]\) (c - a) +
    
31. \!\(\*OverscriptBox["c", "_"]\) (a - b));
    
32. \!\(\*OverscriptBox["Waixin", "_"]\)[a_, b_, c_] := -((a
    
33. \!\(\*OverscriptBox["a", "_"]\) (
    
34. \!\(\*OverscriptBox["b", "_"]\) -
    
35. \!\(\*OverscriptBox["c", "_"]\)) + b
    
36. \!\(\*OverscriptBox["b", "_"]\) (
    
37. \!\(\*OverscriptBox["c", "_"]\) -
    
38. \!\(\*OverscriptBox["a", "_"]\)) + c
    
39. \!\(\*OverscriptBox["c", "_"]\) (
    
40. \!\(\*OverscriptBox["a", "_"]\) -
    
41. \!\(\*OverscriptBox["b", "_"]\)) )/(
    
42. \!\(\*OverscriptBox["a", "_"]\) (b - c) +
    
43. \!\(\*OverscriptBox["b", "_"]\) (c - a) +
    
44. \!\(\*OverscriptBox["c", "_"]\) (a - b)));(*外心公式*)
    
45. kAB = k[a, b]; kAE = kAB v;
    
46. e = Chuizuk[a, kAE, b];
    
47. \!\(\*OverscriptBox["e", "_"]\) =
    
48. \!\(\*OverscriptBox["Chuizuk", "_"]\)[a, kAE, b]; f =
    
49. Chuizuk[a, kAE, c];
    
50. \!\(\*OverscriptBox["f", "_"]\) =
    
51. \!\(\*OverscriptBox["Chuizuk", "_"]\)[a, kAE, c];
    
52. i = Waixin[e, d, f];
    
53. \!\(\*OverscriptBox["i", "_"]\) =
    
54. \!\(\*OverscriptBox["Waixin", "_"]\)[e, d, f];
    
55. 
    
56. \!\(\*OverscriptBox["h", "_"]\) = h =
    
57. \!\(\*OverscriptBox["i", "_"]\) + i - d;(*由kBD=-((h-i)/(d-
    
58. \!\(\*OverscriptBox["i", "_"]\)))=1推出*)
    
59. Simplify[{kAB, kAE}]
    
60. Simplify[{1, e,
    
61. \!\(\*OverscriptBox["e", "_"]\), , f,
    
62. \!\(\*OverscriptBox["f", "_"]\), , i,
    
63. \!\(\*OverscriptBox["i", "_"]\)}]
    
64. Simplify[{2, k[g, h]; k[i, g], k[d, h]; k[i, d], , k[i, g, h],
    
65.   k[i, d, h], k[i, g, h] - k[i, d, h]}]
    
66. Simplify[{k[d, g], k[f, g]}]
    
67. 
    

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denglongshan

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denglongshan

denglongshan 发表于 2022-1-30 20:38
AC＞AB多余，用向量商获得红色标记的结果

这圆是九点圆，并且H就是A在BC的垂足

denglongshan

好题，没有人有兴趣吗

Future_maths

用解析几何的方法可以证明

渣k

Future_maths 发表于 2022-3-20 10:47
用解析几何的方法可以证明

请您给个过程