H是△ABC的垂心,O∈BC,AO⊥BC,圆O以OA为半径,D=AO∩圆O,M=AB∩圆O,.....,证明EPH共线

天山草

H 是 △ABC 的垂心，O ∈ BC，AO ⊥ BC，圆O 以OA 为半径，D = AO ∩ 圆O，M = AB ∩ 圆O，N = AC ∩ 圆O，P = BN ∩ MC，圆O1 = 圆BHC，E = HP ∩ 圆O1。证明：① D ∈ 圆O1;  ② EPH 共线。

天山草

1. Clear["Global`*"];(*令O为坐标原点，A点坐标为 i，BC与实轴重合，AB复斜率为u^2，AC复斜率为v^2*)
   
2. \!\(\*OverscriptBox[\(o\), \(_\)]\) = o = 0; a = I; \!\(\*OverscriptBox[\(a\), \(_\)]\) = -I; d = -I; \!\(\*OverscriptBox[\(d\), \(_\)]\) = I; kAB = u^2;  kAC = v^2;
   
3. k[a_, b_] := (a - b)/(\!\(\*OverscriptBox[\(a\), \(_\)]\) - \!\(\*OverscriptBox[\(b\), \(_\)]\)); (*复斜率定义*)
   
4. (*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：*)
   
5. 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));
   
6. b = Simplify@Jd[kAB, a, 1, o]; \!\(\*OverscriptBox[\(b\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(Jd\), \(_\)]\)[kAB, a, 1, o];
   
7. c = Simplify@Jd[kAC, a, 1, o]; \!\(\*OverscriptBox[\(c\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(Jd\), \(_\)]\)[kAC, a, 1, o];
   
8. m = Simplify@Jd[kAB, a, -kAB, d]; \!\(\*OverscriptBox[\(m\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(Jd\), \(_\)]\)[kAB, a, -kAB, d];
   
9. n = Simplify@Jd[kAC, a, -kAC, d]; \!\(\*OverscriptBox[\(n\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(Jd\), \(_\)]\)[kAC, a, -kAC, d];
   
10. h = Simplify@Jd[-1, o, -kAC, b]; \!\(\*OverscriptBox[\(h\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(Jd\), \(_\)]\)[-1, o, -kAC, b];
    
11. p = Simplify@Jd[k[m, c], c, k[n, b], b]; \!\(\*OverscriptBox[\(p\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(Jd\), \(_\)]\)[k[m, c], c, k[n, b], b];
    
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[b, h, c]; \!\(\*OverscriptBox[\(o1\), \(_\)]\) = Simplify@\!\(\*OverscriptBox[\(WX\), \(_\)]\)[b, h, c];
    
15. W1 = {e, \!\(\*OverscriptBox[\(e\), \(_\)]\)} /. Simplify@Solve[{(o - e) (\!\(\*OverscriptBox[\(o\), \(_\)]\) - \!\(\*OverscriptBox[\(e\), \(_\)]\)) == 1, (o1 - e) (\!\(\*OverscriptBox[\(o1\), \(_\)]\) - \!\(\*OverscriptBox[\(e\), \(_\)]\)) == (o1 - b) (\!\(\*OverscriptBox[\(o1\), \(_\)]\) - \!\(\*OverscriptBox[\(b\), \(_\)]\))}, {e, \!\(\*OverscriptBox[\(e\), \(_\)]\)}] // Flatten;
    
16. d1 = Part[W1, 1]; \!\(\*OverscriptBox[\(d1\), \(_\)]\) = Part[W1, 2]; e = Part[W1, 3]; \!\(\*OverscriptBox[\(e\), \(_\)]\) = Part[W1, 4];Print["D1 = ", d1]; Print["由于 D1 点的坐标为-\[ImaginaryI]，所以D1点与D点重合。"];
    
17. Print["EP的复斜率 kEP = ", Simplify[k[e, p]]];
    
18. Print["EH的复斜率 kEH = ", Simplify[k[e, h]]];
    
19. Print["由于 kEP = kEH，所以 E、P、H 共线。 "];

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