AD是△ABC角A平分线，M 是BC中点。在AB上取点N使MN=MB,过BDN作圆交AD于E,过DCE作圆...

天山草

天山草

此题如果用复平面解析几何做，没有难度。

天山草

程序代码：

1. Clear["Global`*"];(*构图法b=0，c=1，u=\[ExponentialE]^\[ImaginaryI]\[Angle]\
   
2. B ，v=\[ExponentialE]^\[ImaginaryI]\[Angle]C ，kAB=u^2，kAC=1/v^2*)
   
3. \!\(\*OverscriptBox[\(b\), \(_\)]\) = b = 0; \!\(\*OverscriptBox[\(c\), \(_\)]\) = c = 1; a = (u^2 (v^2 - 1))/( u^2 v^2 - 1);
   
4. \!\(\*OverscriptBox[\(a\), \(_\)]\) = (v^2 - 1)/(u^2 v^2 - 1); i = ( u (v - 1))/(u v - 1);
   
5. \!\(\*OverscriptBox[\(i\), \(_\)]\) = (v - 1)/(u v - 1);d = (u (v^2 - 1))/(u^2 v + u (v^2 - 1) - v);  
   
6. \!\(\*OverscriptBox[\(d\), \(_\)]\) = d; m = 1/2; \!\(\*OverscriptBox[\(m\), \(_\)]\) =m;(*此构图下各点的坐标公式事先已算出，以上直接引用，不再推导*)
   
7. n = \[Lambda] a; \!\(\*OverscriptBox[\(n\), \(_\)]\) = \[Lambda] \!\(\*OverscriptBox[\(a\), \(_\)]\);
   
8. Simplify@Solve[{(m - n) (\!\(\*OverscriptBox[\(m\), \(_\)]\) - \!\(\*OverscriptBox[\(n\), \(_\)]\)) == 1/4}, {\[Lambda]}]; \[Lambda] = ((u^2 + 1) (u^2 v^2 - 1))/( 2 u^2 (v^2 - 1)); n = Simplify[\[Lambda] a];
   
9. \!\(\*OverscriptBox[\(n\), \(_\)]\) = Simplify[\[Lambda] \!\(\*OverscriptBox[\(a\), \(_\)]\)];
   
10. (*三角形的外心坐标：*)
    
11. WX[a_, b_, c_] := (a \!\(\*OverscriptBox[\(a\), \(_\)]\) (c - b) + b \!\(\*OverscriptBox[\(b\), \(_\)]\)(a - c) + c \!\(\*OverscriptBox[\(c\), \(_\)]\)(b - a) )/(\!\(\*OverscriptBox[\(a\), \(_\)]\)(c - b) + \!\(\*OverscriptBox[\(b\), \(_\)]\)(a - c) + \!\(\*OverscriptBox[\(c\), \(_\)]\)(b - a));   
    
12. \!\(\*OverscriptBox[\(WX\), \(_\)]\)[a_, b_, c_] := -((a \!\(\*OverscriptBox[\(a\), \(_\)]\)(\!\(\*OverscriptBox[\(c\), \(_\)]\) -
    
13. \!\(\*OverscriptBox[\(b\), \(_\)]\)) + b \!\(\*OverscriptBox[\(b\), \(_\)]\)(\!\(\*OverscriptBox[\(a\), \(_\)]\) - \!\(\*OverscriptBox[\(c\), \(_\)]\)) + c \!\(\*OverscriptBox[\(c\), \(_\)]\)(\!\(\*OverscriptBox[\(b\), \(_\)]\) - \!\(\*OverscriptBox[\(a\), \(_\)]\)))/(
    
14. \!\(\*OverscriptBox[\(a\), \(_\)]\)(c - b) + \!\(\*OverscriptBox[\(b\), \(_\)]\)(a - c) + \!\(\*OverscriptBox[\(c\), \(_\)]\)(b - a)));
    
15. p = Simplify@ WX[b, d, n]; \!\(\*OverscriptBox[\(p\), \(_\)]\) = Simplify@ \!\(\*OverscriptBox[\(WX\), \(_\)]\)[b, d, n];(*过BDN三点的圆的圆心*)
    
16. (*复斜率定义：*) k[a_, b_] := (a - b)/(\!\(\*OverscriptBox[\(a\), \(_\)]\) - \!\(\*OverscriptBox[\(b\), \(_\)]\));
    
17. W1 = {e, \!\(\*OverscriptBox[\(e\), \(_\)]\)} /. Simplify@Solve[{(p - b) (\!\(\*OverscriptBox[\(p\), \(_\)]\) - \!\(\*OverscriptBox[\(b\), \(_\)]\)) == (p - e) (\!\(\*OverscriptBox[\(p\), \(_\)]\) - \!\(\*OverscriptBox[\(e\), \(_\)]\)), k[a, d] == k[a, e], e != d}, {e,
    
18. \!\(\*OverscriptBox[\(e\), \(_\)]\)}] // Flatten ;
    
19. e = Part[W1, 1]; \!\(\*OverscriptBox[\(e\), \(_\)]\) = Part[W1, 2];
    
20. q = Simplify@ WX[e, d, c]; \!\(\*OverscriptBox[\(q\), \(_\)]\) = Simplify@ \!\(\*OverscriptBox[\(WX\), \(_\)]\)[e, d, c];(*过EDC三点的圆的圆心*)
    
21. W2 = {f, \!\(\*OverscriptBox[\(f\), \(_\)]\)} /. Simplify@Solve[{(q - c) (\!\(\*OverscriptBox[\(q\), \(_\)]\) - \!\(\*OverscriptBox[\(c\), \(_\)]\)) == (q - f) (\!\(\*OverscriptBox[\(q\), \(_\)]\) - \!\(\*OverscriptBox[\(f\), \(_\)]\)), k[a, c] == k[a, f], f != c}, {f, \!\(\*OverscriptBox[\(f\), \(_\)]\)}] // Flatten ;
    
22. f = Part[W2, 1]; \!\(\*OverscriptBox[\(f\), \(_\)]\) = Part[W2, 2];
    
23. Print["测试 MF = MC 是否成立："];
    
24. Simplify[(m - f) (\!\(\*OverscriptBox[\(m\), \(_\)]\) - \!\(\*OverscriptBox[\(f\), \(_\)]\)) == (m - c) (\!\(\*OverscriptBox[\(m\), \(_\)]\) - \!\(\*OverscriptBox[\(c\), \(_\)]\))]
    
25. Print["测试 ME 是否为 \[Angle]FMN 的平分线："];
    
26. Simplify[k[m, e]^2 == k[m, f] k[m, n]]

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