I,O,IB,IC是ΔABC内心外心旁心，E=BI∩AC,F=CI∩AB，P,Q=EF∩⊙O，证：IB,IC在圆PIQ上

天山草

I 是 △ABC 的内心，O 是外心，E = BI∩AC，F = CI∩AB，直线 FE 与圆O 交于 P、Q。
\(I_B\)、\(I_C\) 是旁切圆圆心，证明：① 圆PIQ 的半径等于圆O 半径的二倍。  ② 这两个旁切圆心在圆PIQ 上。

天山草

程序代码：

1. Clear["Global`*"]; (*令△ABC的外接圆为单位圆O，BC边平行于实轴，AB、AC的复斜率分别为 u^2、v^2 *)
   
2. o = 0; \!\(\*OverscriptBox[\(o\), \(_\)]\) = 0; a = I u v;   \!\(\*OverscriptBox[\(a\), \(_\)]\) = 1/(I u v); b = (I u)/v;  
   
3. \!\(\*OverscriptBox[\(b\), \(_\)]\) = v/(I u);  c = (I v)/u;  \!\(\*OverscriptBox[\(c\), \(_\)]\) = u/(I v);
   
4. i = I (u + v - 1); \!\(\*OverscriptBox[\(i\), \(_\)]\) = -I (1/u + 1/v - 1);
   
5. k[a_, b_] := (a - b)/(\!\(\*OverscriptBox[\(a\), \(_\)]\) - \!\(\*OverscriptBox[\(b\), \(_\)]\)); (*复斜率定义*)
   
6. (*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：*)
   
7. Jd[k1_, a1_, k2_, a2_] := -((k2 (a1 - k1 \!\(\*OverscriptBox[\(a1\), \(_\)]\)) - k1 (a2 - k2 \!\(\*OverscriptBox[\(a2\), \(_\)]\)))/(k1 - k2));\!\(\*OverscriptBox[\(Jd\), \(_\)]\)[k1_, a1_, k2_, a2_] := -((a1 - k1 \!\(\*OverscriptBox[\(a1\), \(_\)]\) - (a2 - k2 \!\(\*OverscriptBox[\(a2\), \(_\)]\)))/(k1 - k2));
   
8. e = Simplify@Jd[k[a, c], a, k[b, i], b]; \!\(\*OverscriptBox[\(e\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(Jd\), \(_\)]\)[k[a, c], a, k[b, i], b];
   
9. f = Simplify@Jd[k[a, b], a, k[c, i], c]; \!\(\*OverscriptBox[\(f\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(Jd\), \(_\)]\)[k[a, b], a, k[c, i], c];
   
10. W1 = {z, \!\(\*OverscriptBox[\(z\), \(_\)]\)} /. Simplify@Solve[{(o - z) (\!\(\*OverscriptBox[\(o\), \(_\)]\) - \!\(\*OverscriptBox[\(z\), \(_\)]\)) == 1, k[f, e] == k[e, z]}, {z, \!\(\*OverscriptBox[\(z\), \(_\)]\)}] // Flatten;
    
11. p = Part[W1, 1]; \!\(\*OverscriptBox[\(p\), \(_\)]\) = Part[W1, 2]; q = Part[W1, 3]; \!\(\*OverscriptBox[\(q\), \(_\)]\) = Part[W1, 4];
    
12. 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));(*三角形 ABC 的外心坐标：*)   
    
13. \!\(\*OverscriptBox[\(WX\), \(_\)]\)[a_, b_, c_] := (\!\(\*OverscriptBox[\(a\), \(_\)]\) \!\(\*OverscriptBox[\(b\), \(_\)]\) (a - b) + \!\(\*OverscriptBox[\(b\), \(_\)]\) \!\(\*OverscriptBox[\(c\), \(_\)]\) (b - c) + \!\(\*OverscriptBox[\(c\), \(_\)]\) \!\(\*OverscriptBox[\(a\), \(_\)]\) (c - a))/(\!\(\*OverscriptBox[\(a\), \(_\)]\) (c - b) + \!\(\*OverscriptBox[\(b\), \(_\)]\) (a - c) + \!\(\*OverscriptBox[\(c\), \(_\)]\) (b - a));
    
14. o1 = Simplify@WX[i, p, q]; \!\(\*OverscriptBox[\(o1\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(WX\), \(_\)]\)[i, p, q];
    
15. Print["O1 圆的半径 = ", Simplify[Sqrt[(o1 - i) (\!\(\*OverscriptBox[\(o1\), \(_\)]\) - \!\(\*OverscriptBox[\(i\), \(_\)]\))]]];
    
16. Print["由于圆O1的半径为 2，圆O的半径为 1，所以圆O1半径等于圆O半径的二倍。"];
    
17. BC = Simplify[Sqrt[(b - c) (\!\(\*OverscriptBox[\(b\), \(_\)]\) - \!\(\*OverscriptBox[\(c\), \(_\)]\))]]; BC = ( I (v^2 - u^2))/(u v);
    
18. AC = Simplify[Sqrt[(a - c) (\!\(\*OverscriptBox[\(a\), \(_\)]\) - \!\(\*OverscriptBox[\(c\), \(_\)]\))]]; AC = (I (1 - u^2))/u;
    
19. AB = Simplify[Sqrt[(a - b) (\!\(\*OverscriptBox[\(a\), \(_\)]\) - \!\(\*OverscriptBox[\(b\), \(_\)]\))]]; AB = (I (v^2 - 1))/v;
    
20. Subscript[I, B] = Simplify[(BC a - AC b + AB c)/(BC - AC + AB)];
    
21. \!\(\*OverscriptBox[SubscriptBox[\(I\), \(B\)], \(_\)]\) = Simplify[(BC \!\(\*OverscriptBox[\(a\), \(_\)]\) - AC \!\(\*OverscriptBox[\(b\), \(_\)]\) + AB \!\(\*OverscriptBox[\(c\), \(_\)]\))/(BC - AC + AB)];
    
22. Subscript[I, C] = Simplify[(BC a + AC b - AB c)/(BC + AC - AB)]; \!\(\*OverscriptBox[SubscriptBox[\(I\), \(C\)], \(_\)]\) = Simplify[(BC \!\(\*OverscriptBox[\(a\), \(_\)]\) + AC \!\(\*OverscriptBox[\(b\), \(_\)]\) - AB \!\(\*OverscriptBox[\(c\), \(_\)]\))/(BC + AC - AB)];
    
23. Print["p = ", p]; Print["q = ", q]; Print["o1 = ", o1]; Print["\!\(\*SubscriptBox[\(I\), \(B\)]\) = ", Subscript[I, B]]; \
    
24. Print["\!\(\*SubscriptBox[\(I\), \(C\)]\) = ", Subscript[I, C]];
    
25. Print["\!\(\*SubscriptBox[\(O1I\), \(B\)]\) = ",  Simplify[Sqrt[(o1 - Subscript[I, B]) (\!\(\*OverscriptBox[\(o1\), \(_\)]\) - \!\(\*OverscriptBox[SubscriptBox[\(I\), \(B\)], \(_\)]\))]]];
    
26. Print["\!\(\*SubscriptBox[\(O1I\), \(C\)]\) = ",  Simplify[Sqrt[(o1 - Subscript[I, C]) (\!\(\*OverscriptBox[\(o1\), \(_\)]\) - \!\(\*OverscriptBox[SubscriptBox[\(I\), \(C\)], \(_\)]\))]]];
    
27. Print["由于 \!\(\*SubscriptBox[\(O1I\), \(B\)]\) = 2，所以 \!\(\*SubscriptBox[\(I\), \(B\)]\) 在 O1 圆上。"];
    
28. Print["由于 \!\(\*SubscriptBox[\(O1I\), \(C\)]\) = 2，所以 \!\(\*SubscriptBox[\(I\), \(C\)]\) 在 O1 圆上。"];

复制代码

程序运行结果：

天山草

有些意外的是，P 和 Q 的坐标那么复杂，还含有根式，但并不影响后续点坐标的简单性且不含根式。