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dlsh

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百度几何吧转载https://tieba.baidu.com/p/7451652229

dlsh

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。

dlsh

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. 
   
3. 
   
4. \!\(\*OverscriptBox["a", "_"]\) = 1/a;
   
5. \!\(\*OverscriptBox["b", "_"]\) = 1/b;
   
6. \!\(\*OverscriptBox["c", "_"]\) = 1/c;(*
   
7. \!\(\*OverscriptBox["p", "_"]\)=-((-a+2 b-c-p)/(a c ))*);
   
8. 
   
9. k[a_, b_] := (a - b)/(
   
10. \!\(\*OverscriptBox["a", "_"]\) -
    
11. \!\(\*OverscriptBox["b", "_"]\));
    
12. \!\(\*OverscriptBox["k", "_"]\)[a_, b_] := 1/k[a, b];(*复斜率定义*)
    
13. 
    
14. \!\(\*OverscriptBox["Jd", "_"]\)[k1_, a1_, k2_, a2_] := -((a1 - k1
    
15. \!\(\*OverscriptBox["a1", "_"]\) - (a2 - k2
    
16. \!\(\*OverscriptBox["a2", "_"]\)))/(
    
17.    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*)
    
18. Jd[k1_, a1_, k2_, a2_] := -((k2 (a1 - k1
    
19. \!\(\*OverscriptBox["a1", "_"]\)) - k1 (a2 - k2
    
20. \!\(\*OverscriptBox["a2", "_"]\)))/(k1 - k2));
    
21. Chuizu[a_, b_, p_] := (
    
22. \!\(\*OverscriptBox["a", "_"]\) b - a
    
23. \!\(\*OverscriptBox["b", "_"]\) + p (
    
24. \!\(\*OverscriptBox["a", "_"]\) -
    
25. \!\(\*OverscriptBox["b", "_"]\)) +
    
26. \!\(\*OverscriptBox["p", "_"]\) (a - b))/(2 (
    
27. \!\(\*OverscriptBox["a", "_"]\) -
    
28. \!\(\*OverscriptBox["b", "_"]\)));(*=(1/2)[p+(
    
29. \!\(\*OverscriptBox["a", "_"]\)b-a
    
30. \!\(\*OverscriptBox["b", "_"]\)+
    
31. \!\(\*OverscriptBox["p", "_"]\)(a-b))/(
    
32. \!\(\*OverscriptBox["a", "_"]\)-
    
33. \!\(\*OverscriptBox["b", "_"]\))]P到直线AB的垂足*)
    
34. 
    
35. \!\(\*OverscriptBox["Chuizu", "_"]\)[a_, b_, p_] := (a
    
36. \!\(\*OverscriptBox["b", "_"]\) -
    
37. \!\(\*OverscriptBox["a", "_"]\) b +
    
38. \!\(\*OverscriptBox["p", "_"]\) (a - b) + p (
    
39. \!\(\*OverscriptBox["a", "_"]\) -
    
40. \!\(\*OverscriptBox["b", "_"]\)))/(2 (a - b));
    
41. FourPoint[a_, b_, c_, d_] := ((
    
42. \!\(\*OverscriptBox["c", "_"]\) d - c
    
43. \!\(\*OverscriptBox["d", "_"]\)) (a - b) - (
    
44. \!\(\*OverscriptBox["a", "_"]\) b - a
    
45. \!\(\*OverscriptBox["b", "_"]\)) (c - d))/((a - b) (
    
46. \!\(\*OverscriptBox["c", "_"]\) -
    
47. \!\(\*OverscriptBox["d", "_"]\)) - (
    
48. \!\(\*OverscriptBox["a", "_"]\) -
    
49. \!\(\*OverscriptBox["b", "_"]\)) (c - d));(*过两点A和B、C和D的交点*)
    
50. 
    
51. \!\(\*OverscriptBox["FourPoint", "_"]\)[a_, b_, c_, d_] := -(((c
    
52. \!\(\*OverscriptBox["d", "_"]\) -
    
53. \!\(\*OverscriptBox["c", "_"]\) d) (
    
54. \!\(\*OverscriptBox["a", "_"]\) -
    
55. \!\(\*OverscriptBox["b", "_"]\)) - ( a
    
56. \!\(\*OverscriptBox["b", "_"]\) -
    
57. \!\(\*OverscriptBox["a", "_"]\) b) (
    
58. \!\(\*OverscriptBox["c", "_"]\) -
    
59. \!\(\*OverscriptBox["d", "_"]\)))/((a - b) (
    
60. \!\(\*OverscriptBox["c", "_"]\) -
    
61. \!\(\*OverscriptBox["d", "_"]\)) - (
    
62. \!\(\*OverscriptBox["a", "_"]\) -
    
63. \!\(\*OverscriptBox["b", "_"]\)) (c - d)));
    
64. f = Chuizu[a, b, p];
    
65. \!\(\*OverscriptBox["f", "_"]\) =
    
66. \!\(\*OverscriptBox["Chuizu", "_"]\)[a , b, p]; e = Chuizu[a , c, p];
    
67. \!\(\*OverscriptBox["e", "_"]\) =
    
68. \!\(\*OverscriptBox["Chuizu", "_"]\)[a , c, p]; d = Chuizu[b, c, p];
    
69. \!\(\*OverscriptBox["d", "_"]\) =
    
70. \!\(\*OverscriptBox["Chuizu", "_"]\)[c, b, p];
    
71. r1 = FourPoint[f, c, b, e];
    
72. \!\(\*OverscriptBox["r1", "_"]\) =
    
73. \!\(\*OverscriptBox["FourPoint", "_"]\)[f, c, b, e]; r =
    
74. FourPoint[a, d, b, e];
    
75. \!\(\*OverscriptBox["r", "_"]\) =
    
76. \!\(\*OverscriptBox["FourPoint", "_"]\)[a, d, b, e];
    
77. u = Jd[-k[c, p], b, -a c, a];
    
78. \!\(\*OverscriptBox["u", "_"]\) =
    
79. \!\(\*OverscriptBox["Jd", "_"]\)[-k[c, p], b, -a c, a]; v =
    
80. Jd[-k[b, p], c, -a b, a];
    
81. \!\(\*OverscriptBox["v", "_"]\) =
    
82. \!\(\*OverscriptBox["Jd", "_"]\)[-k[b, p], c, -a b, a];
    
83. t = FourPoint[b, c, u, v];
    
84. \!\(\*OverscriptBox["t", "_"]\) =
    
85. \!\(\*OverscriptBox["FourPoint", "_"]\)[b, c, u, v];
    
86. 
    
87. Simplify[{r, r1, , r1 - r}](*验证Ceva点*)
    
88. Factor[{1, r, r1, , r1 - r}]
    
89. Simplify[{2, u,
    
90. \!\(\*OverscriptBox["u", "_"]\), , v, v, , t, t, , k[a, t], k[p, r], ,
    
91.    k[a, t] - k[p, r]}](*k[p,r],k[p,r]+k[a,t]*)
    
92. Factor[{2, u,
    
93. \!\(\*OverscriptBox["u", "_"]\), , v, v, , t, t, , k[a, t],
    
94.   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。