直线交点在圆上，实在不知道程序错在哪里

dlsh

1. 
   
2. \!\(\*OverscriptBox["o", "_"]\) = o = 0;
   
3. \!\(\*OverscriptBox["c", "_"]\) = c = 1;
   
4. \!\(\*OverscriptBox["a", "_"]\) = 1/a;
   
5. \!\(\*OverscriptBox["b", "_"]\) = 1/b;
   
6. \!\(\*OverscriptBox["t", "_"]\) = 1/t; b = t^2; h = a + b + c;
   
7. \!\(\*OverscriptBox["h", "_"]\) =
   
8. \!\(\*OverscriptBox["a", "_"]\) +
   
9. \!\(\*OverscriptBox["b", "_"]\) +
   
10. \!\(\*OverscriptBox["c", "_"]\); m = (a + h)/2;
    
11. \!\(\*OverscriptBox["m", "_"]\) = (
    
12. \!\(\*OverscriptBox["a", "_"]\) +
    
13. \!\(\*OverscriptBox["h", "_"]\))/2;
    
14. k[a_, b_] := (a - b)/(
    
15. \!\(\*OverscriptBox["a", "_"]\) -
    
16. \!\(\*OverscriptBox["b", "_"]\));
    
17. \!\(\*OverscriptBox["k", "_"]\)[a_, b_] := 1/k[a, b];(*复斜率定义*)
    
18. 
    
19. \!\(\*OverscriptBox["Jd", "_"]\)[k1_, a1_, k2_, a2_] := -((a1 - k1
    
20. \!\(\*OverscriptBox["a1", "_"]\) - (a2 - k2
    
21. \!\(\*OverscriptBox["a2", "_"]\)))/(k1 - k2));
    
22. (*复斜率等于k1,过点A1与复斜率等于k2,过点A2的直线交点*)
    
23. Jd[k1_, a1_, k2_, a2_] := -((k2 (a1 - k1
    
24. \!\(\*OverscriptBox["a1", "_"]\)) - k1 (a2 - k2
    
25. \!\(\*OverscriptBox["a2", "_"]\)))/(k1 - k2));
    
26. FourPoint[a_, b_, c_, d_] := ((
    
27. \!\(\*OverscriptBox["c", "_"]\) d - c
    
28. \!\(\*OverscriptBox["d", "_"]\)) (a - b) - (
    
29. \!\(\*OverscriptBox["a", "_"]\) b - a
    
30. \!\(\*OverscriptBox["b", "_"]\)) (c - d))/((a - b) (
    
31. \!\(\*OverscriptBox["c", "_"]\) -
    
32. \!\(\*OverscriptBox["d", "_"]\)) - (
    
33. \!\(\*OverscriptBox["a", "_"]\) -
    
34. \!\(\*OverscriptBox["b", "_"]\)) (c - d));(*过两点A和B、C和D的交点*)
    
35. 
    
36. \!\(\*OverscriptBox["FourPoint", "_"]\)[a_, b_, c_, d_] := -(((c
    
37. \!\(\*OverscriptBox["d", "_"]\) -
    
38. \!\(\*OverscriptBox["c", "_"]\) d) (
    
39. \!\(\*OverscriptBox["a", "_"]\) -
    
40. \!\(\*OverscriptBox["b", "_"]\)) - ( a
    
41. \!\(\*OverscriptBox["b", "_"]\) -
    
42. \!\(\*OverscriptBox["a", "_"]\) b) (
    
43. \!\(\*OverscriptBox["c", "_"]\) -
    
44. \!\(\*OverscriptBox["d", "_"]\)))/((a - b) (
    
45. \!\(\*OverscriptBox["c", "_"]\) -
    
46. \!\(\*OverscriptBox["d", "_"]\)) - (
    
47. \!\(\*OverscriptBox["a", "_"]\) -
    
48. \!\(\*OverscriptBox["b", "_"]\)) (c - d)));
    
49. Duichengdian[a_, b_, p_] := (
    
50. \!\(\*OverscriptBox["a", "_"]\) b - a
    
51. \!\(\*OverscriptBox["b", "_"]\) +
    
52. \!\(\*OverscriptBox["p", "_"]\) (a - b))/(
    
53. \!\(\*OverscriptBox["a", "_"]\) -
    
54. \!\(\*OverscriptBox["b", "_"]\));(*P关于AB的对称点*)
    
55. 
    
56. \!\(\*OverscriptBox["Duichengdian", "_"]\)[a_, b_, p_] := (a
    
57. \!\(\*OverscriptBox["b", "_"]\) -
    
58. \!\(\*OverscriptBox["a", "_"]\) b + p (
    
59. \!\(\*OverscriptBox["a", "_"]\) -
    
60. \!\(\*OverscriptBox["b", "_"]\)))/(a - b);
    
61. o1 = Jd[a b, m, -a t, a];
    
62. \!\(\*OverscriptBox["o1", "_"]\) =
    
63. \!\(\*OverscriptBox["Jd", "_"]\)[a b, m, -a t, a];
    
64. d = a b (
    
65. \!\(\*OverscriptBox["a", "_"]\) -
    
66. \!\(\*OverscriptBox["o1", "_"]\)) + o1;
    
67. \!\(\*OverscriptBox["d", "_"]\) =
    
68. \!\(\*OverscriptBox["a", "_"]\)
    
69. \!\(\*OverscriptBox["b", "_"]\) (a - o1) +
    
70. \!\(\*OverscriptBox["o1", "_"]\); e = a c (
    
71. \!\(\*OverscriptBox["a", "_"]\) -
    
72. \!\(\*OverscriptBox["o1", "_"]\)) + o1;
    
73. \!\(\*OverscriptBox["e", "_"]\) =
    
74. \!\(\*OverscriptBox["a", "_"]\)
    
75. \!\(\*OverscriptBox["c", "_"]\) (a - o1) +
    
76. \!\(\*OverscriptBox["o1", "_"]\);
    
77. x = Duichengdian[o, o1, a];
    
78. \!\(\*OverscriptBox["x", "_"]\) =
    
79. \!\(\*OverscriptBox["Duichengdian", "_"]\)[o, o1, a];
    
80. q = -(k[x, d]/x);
    
81. \!\(\*OverscriptBox["q", "_"]\) = 1/q; p = -(k[x, e]/x);
    
82. \!\(\*OverscriptBox["p", "_"]\) = 1/p;
    
83. s = FourPoint[q, e, p, d];
    
84. \!\(\*OverscriptBox["s", "_"]\) =
    
85. \!\(\*OverscriptBox["FourPoint", "_"]\)[q, e, p, d];
    
86. s1 = -k[q, e] (
    
87. \!\(\*OverscriptBox["e", "_"]\) -
    
88. \!\(\*OverscriptBox["o1", "_"]\)) + o1; s2 = -k[p, d] (
    
89. \!\(\*OverscriptBox["d", "_"]\) -
    
90. \!\(\*OverscriptBox["o1", "_"]\)) + o1;
    
91. Simplify[{1, o1,
    
92. \!\(\*OverscriptBox["o1", "_"]\), , d,
    
93. \!\(\*OverscriptBox[
    
94. RowBox[{"d", "\[IndentingNewLine]"}], "_"]\), e,
    
95. \!\(\*OverscriptBox[
    
96. RowBox[{"e", "\[IndentingNewLine]"}], "_"]\), , x,
    
97. \!\(\*OverscriptBox["x", "_"]\)}]
    
98. Simplify[{2, p,
    
99. \!\(\*OverscriptBox["p", "_"]\), , q,
    
100. \!\(\*OverscriptBox[
     
101. RowBox[{"q", "\[IndentingNewLine]"}], "_"]\), , s,
     
102. \!\(\*OverscriptBox["s", "_"]\)}]
     
103. Simplify[{3, k[s, e], -((s - o1)/(
     
104. \!\(\*OverscriptBox["e", "_"]\) -
     
105. \!\(\*OverscriptBox["o1", "_"]\))), , k[s, d], k[p, d], -((s - o1)/(
     
106. \!\(\*OverscriptBox["d", "_"]\) -
     
107. \!\(\*OverscriptBox["o1", "_"]\))), -((d - o1)/(
     
108. \!\(\*OverscriptBox["s", "_"]\) -
     
109. \!\(\*OverscriptBox["o1", "_"]\))), , -((s - o1)/(
     
110. \!\(\*OverscriptBox["d", "_"]\) -
     
111. \!\(\*OverscriptBox["o1", "_"]\))) + (d - o1)/(
     
112. \!\(\*OverscriptBox["s", "_"]\) -
     
113. \!\(\*OverscriptBox["o1", "_"]\))}](*验证S在圆O1上*)
     
114. Simplify[{(s - o1) (
     
115. \!\(\*OverscriptBox["s", "_"]\) -
     
116. \!\(\*OverscriptBox["o1", "_"]\)), , (x - o1) (
     
117. \!\(\*OverscriptBox["x", "_"]\) -
     
118. \!\(\*OverscriptBox["o1", "_"]\)), (o1 - a) (
     
119. \!\(\*OverscriptBox["o1", "_"]\) -
     
120. \!\(\*OverscriptBox["a", "_"]\)), , (e - o1) (
     
121. \!\(\*OverscriptBox["e", "_"]\) -
     
122. \!\(\*OverscriptBox["o1", "_"]\)), (d - o1) (
     
123. \!\(\*OverscriptBox["d", "_"]\) -
     
124. \!\(\*OverscriptBox["o1", "_"]\)), x
     
125. \!\(\*OverscriptBox["x", "_"]\)}]
     
126. Simplify[{(s - o1) (
     
127. \!\(\*OverscriptBox["s", "_"]\) -
     
128. \!\(\*OverscriptBox["o1", "_"]\)) - (x - o1) (
     
129. \!\(\*OverscriptBox["x", "_"]\) -
     
130. \!\(\*OverscriptBox["o1", "_"]\))}]
     
131. Simplify[{k[a, d], k[a, e], -((a - o1)/(
     
132. \!\(\*OverscriptBox["d", "_"]\) -
     
133. \!\(\*OverscriptBox["o1", "_"]\)))}]
     
134. Simplify[{s1, s2, , s1 - s2}]
     
135. (*验证S在圆O1上*)
     
136. 
     

复制代码

最后结果错误

最后的S点错误，三次都不同，不知道错在哪里

TSC999

你的运行结果中没有共轭字符出现，复制你的代码在我的 mathematica 9.0 上运行，运行结果中就有共轭字符出现（简化不彻底）。不知是何原因。
你在程序最前面加上一条清除所有变量的指令，看看每次运行情况是不是都一样。

TSC999

为查找问题，把程序弄到最简，如下图。
这个程序首次运行有问题 (结果中有共轭复数)，但是第二次、第三次、.....  运行就一直正常了。
估计问题出在图中的蓝色字符上。


下图是首次运行结果与再次运行结果的比较：

TSC999

A 点、P 点的位置在哪里？ 程序中只表明它们都在单位圆上，但是并没有具体位置的定义。这也许是导致 s 出错的原因？

s  的结果是稳定的，并没有三次运行的结果都不相同。S  点位于 O1 圆上的判定依据是什么？

dlsh

如果S在O1上，S与A到O1的距离平方相等 或者-(s-o1)/(e'-o1')=kqe,-(d-o1)/(s'-o1')=kpd,为简化用撇号代替共轭

TSC999

为了查找程序中的计算错误，画出下图。

给定 A、T 的坐标，其它点的坐标都可按程序计算出来。在图中实测各点坐标，发现有的点 (例如 D、E 、X、O1、S 点) 计算与实测不符。

AD、AE  长度的计算值也与实测值不一样。



以下是换一种方法计算 O1、D、E 、X、P、Q、S 各点的坐标以及 AD、AE 的长度，结果全都与实测值相符合：

o1= - 0.4280947 + I 0.2345752
d = - 0.587453 - I 0 .26846
e =   0.092019 + I 0 .323577
x = - 0.950037 + I 0 .312138
p =   0.942956 + I 0 .332918
q = - 0.136446 - I 0 .990647
s =   0.0313717 - I 0 .0252913
AD = 0.920377
AE = 0.920111

终于发现原程序中的错误是求 o1 的那条语句，应改为  o1=Simplify[Jd[-k[a, h] , m , k[a, t], a ]];  原程序中的  o1=Simplify[Jd[a b , m , -a t, a ]]; 是不对的。这个错误不但造成计算结果不对，并且发生程序开头不能加上清除所有变量语句的奇怪现象。
改正这个错误后，一切问题都解决了。下页是更正后的正确程序代码。

TSC999

1. Clear["Global`*"]; (*设三角形ABC的外接圆为单位圆，C 点在正 x 轴上*)
   
2. 
   
3. \!\(\*OverscriptBox[\(o\), \(_\)]\) = o = 0;
   
4. \!\(\*OverscriptBox[\(c\), \(_\)]\) = c = 1;
   
5. 
   
6. \!\(\*OverscriptBox[\(a\), \(_\)]\) = 1/a; b = t^2;
   
7. \!\(\*OverscriptBox[\(b\), \(_\)]\) = 1/b;
   
8. \!\(\*OverscriptBox[\(t\), \(_\)]\) = 1/t;
   
9. \!\(\*OverscriptBox[\(p\), \(_\)]\) = 1/p;
   
10. h = a + b + c(*当三角形外心在坐标原点时此式成立*);
    
11. 
    
12. \!\(\*OverscriptBox[\(h\), \(_\)]\) =
    
13. \!\(\*OverscriptBox[\(a\), \(_\)]\) +
    
14. \!\(\*OverscriptBox[\(b\), \(_\)]\) +
    
15. \!\(\*OverscriptBox[\(c\), \(_\)]\); m = (a + h)/2;(* M是A、H的中点*)
    
16. \!\(\*OverscriptBox[\(m\), \(_\)]\) = (
    
17. \!\(\*OverscriptBox[\(a\), \(_\)]\) +
    
18. \!\(\*OverscriptBox[\(h\), \(_\)]\))/2;
    
19. k[a_, b_] := (a - b)/(
    
20. \!\(\*OverscriptBox[\(a\), \(_\)]\) -
    
21. \!\(\*OverscriptBox[\(b\), \(_\)]\));
    
22. \!\(\*OverscriptBox[\(k\), \(_\)]\)[a_, b_] := 1/k[a, b];(*复斜率定义*)
    
23. Jd[k1_, a1_, k2_, a2_] := -((k2 (a1 - k1
    
24. \!\(\*OverscriptBox[\(a1\), \(_\)]\)) - k1 (a2 - k2
    
25. \!\(\*OverscriptBox[\(a2\), \(_\)]\)))/(k1 - k2));
    
26. 
    
27. \!\(\*OverscriptBox[\(Jd\), \(_\)]\)[k1_, a1_, k2_, a2_] := -((a1 - k1
    
28. \!\(\*OverscriptBox[\(a1\), \(_\)]\) - (a2 - k2
    
29. \!\(\*OverscriptBox[\(a2\), \(_\)]\)))/(k1 - k2));
    
30. (*复斜率等于k1且经过A1点的直线，与复斜率等于k2且经过A2点的直线，两直线的交点*)
    
31. FourPoint[a_, b_, c_, d_] := ((
    
32. \!\(\*OverscriptBox[\(c\), \(_\)]\) d - c
    
33. \!\(\*OverscriptBox[\(d\), \(_\)]\)) (a - b) - (
    
34. \!\(\*OverscriptBox[\(a\), \(_\)]\) b - a
    
35. \!\(\*OverscriptBox[\(b\), \(_\)]\)) (c - d))/((a - b) (
    
36. \!\(\*OverscriptBox[\(c\), \(_\)]\) -
    
37. \!\(\*OverscriptBox[\(d\), \(_\)]\)) - (
    
38. \!\(\*OverscriptBox[\(a\), \(_\)]\) -
    
39. \!\(\*OverscriptBox[\(b\), \(_\)]\)) (c - d));
    
40. 
    
41. \!\(\*OverscriptBox[\(FourPoint\), \(_\)]\)[a_, b_, c_, d_] := -(((c
    
42. \!\(\*OverscriptBox[\(d\), \(_\)]\) -
    
43. \!\(\*OverscriptBox[\(c\), \(_\)]\) d) (
    
44. \!\(\*OverscriptBox[\(a\), \(_\)]\) -
    
45. \!\(\*OverscriptBox[\(b\), \(_\)]\)) - ( a
    
46. \!\(\*OverscriptBox[\(b\), \(_\)]\) -
    
47. \!\(\*OverscriptBox[\(a\), \(_\)]\) b) (
    
48. \!\(\*OverscriptBox[\(c\), \(_\)]\) -
    
49. \!\(\*OverscriptBox[\(d\), \(_\)]\)))/((a - b) (
    
50. \!\(\*OverscriptBox[\(c\), \(_\)]\) -
    
51. \!\(\*OverscriptBox[\(d\), \(_\)]\)) - (
    
52. \!\(\*OverscriptBox[\(a\), \(_\)]\) -
    
53. \!\(\*OverscriptBox[\(b\), \(_\)]\)) (c - d)));
    
54. (*过A、B点的直线与过C、D点的直线的交点*)
    
55. Duichengdian[a_, b_, p_] := (
    
56. \!\(\*OverscriptBox[\(a\), \(_\)]\) b - a
    
57. \!\(\*OverscriptBox[\(b\), \(_\)]\) +
    
58. \!\(\*OverscriptBox[\(p\), \(_\)]\) (a - b))/(
    
59. \!\(\*OverscriptBox[\(a\), \(_\)]\) -
    
60. \!\(\*OverscriptBox[\(b\), \(_\)]\));(*P关于直线AB的镜像点*)
    
61. 
    
62. \!\(\*OverscriptBox[\(Duichengdian\), \(_\)]\)[a_, b_, p_] := (a
    
63. \!\(\*OverscriptBox[\(b\), \(_\)]\) -
    
64. \!\(\*OverscriptBox[\(a\), \(_\)]\) b + p (
    
65. \!\(\*OverscriptBox[\(a\), \(_\)]\) -
    
66. \!\(\*OverscriptBox[\(b\), \(_\)]\)))/(a - b);
    
67. (*右边语句错了，不但计算结果不对，并且发生程序开头不能加上清除所有变量语句的奇怪现象o1 =Simplify[ Jd[a b, m, \
    
68. -a t, a]];  Overscript[o1, _] =Simplify[Overscript[Jd, _][a b, m, -a \
    
69. t, a]];*)
    
70. o1 = Simplify[ Jd[-k[a, h], m, k[a, t], a]];
    
71. 
    
72. \!\(\*OverscriptBox[\(o1\), \(_\)]\) = Simplify[
    
73. \!\(\*OverscriptBox[\(Jd\), \(_\)]\)[-k[a, h], m, k[a, t], a]];
    
74. Print["o1=", o1, ",    \!\(\*OverscriptBox[\(o1\), \(_\)]\)=",
    
75. \!\(\*OverscriptBox[\(o1\), \(_\)]\)];
    
76. d = Simplify[ a b (
    
77. \!\(\*OverscriptBox[\(a\), \(_\)]\) -
    
78. \!\(\*OverscriptBox[\(o1\), \(_\)]\)) + o1];
    
79. 
    
80. \!\(\*OverscriptBox[\(d\), \(_\)]\) = Simplify[
    
81. \!\(\*OverscriptBox[\(a\), \(_\)]\)
    
82. \!\(\*OverscriptBox[\(b\), \(_\)]\) (a - o1) +
    
83. \!\(\*OverscriptBox[\(o1\), \(_\)]\)];
    
84. Print["d=", d, ",    \!\(\*OverscriptBox[\(d\), \(_\)]\)=",
    
85. \!\(\*OverscriptBox[\(d\), \(_\)]\)];
    
86. e = Simplify[ a c (
    
87. \!\(\*OverscriptBox[\(a\), \(_\)]\) -
    
88. \!\(\*OverscriptBox[\(o1\), \(_\)]\)) + o1];
    
89. 
    
90. \!\(\*OverscriptBox[\(e\), \(_\)]\) = Simplify[
    
91. \!\(\*OverscriptBox[\(a\), \(_\)]\)
    
92. \!\(\*OverscriptBox[\(c\), \(_\)]\) (a - o1) +
    
93. \!\(\*OverscriptBox[\(o1\), \(_\)]\)];
    
94. Print["e=", e, ",    \!\(\*OverscriptBox[\(e\), \(_\)]\)=",
    
95. \!\(\*OverscriptBox[\(e\), \(_\)]\)];
    
96. x = Simplify[Duichengdian[o, o1, a]];
    
97. \!\(\*OverscriptBox[\(x\), \(_\)]\) = Simplify[
    
98. \!\(\*OverscriptBox[\(Duichengdian\), \(_\)]\)[o, o1, a]];
    
99. Print["x=", x, ",    \!\(\*OverscriptBox[\(x\), \(_\)]\)=",
    
100. \!\(\*OverscriptBox[\(x\), \(_\)]\)];
     
101. q = Simplify[-(k[x, d]/x)];
     
102. \!\(\*OverscriptBox[\(q\), \(_\)]\) = Simplify[1/q]; p =
     
103. Simplify[-(k[x, e]/x)];
     
104. \!\(\*OverscriptBox[\(p\), \(_\)]\) = Simplify[1/p];
     
105. Print["p=", p, ",    \!\(\*OverscriptBox[\(p\), \(_\)]\)=",
     
106. \!\(\*OverscriptBox[\(p\), \(_\)]\)];
     
107. Print["q=", q, ",    \!\(\*OverscriptBox[\(q\), \(_\)]\)=",
     
108. \!\(\*OverscriptBox[\(q\), \(_\)]\)];
     
109. s = Simplify[FourPoint[q, e, p, d]];
     
110. \!\(\*OverscriptBox[\(s\), \(_\)]\) = Simplify[
     
111. \!\(\*OverscriptBox[\(FourPoint\), \(_\)]\)[q, e, p, d]];
     
112. s1 = Simplify[-k[q, e] (
     
113. \!\(\*OverscriptBox[\(e\), \(_\)]\) -
     
114. \!\(\*OverscriptBox[\(o1\), \(_\)]\)) + o1]; s2 = Simplify[-k[p, d] (
     
115. \!\(\*OverscriptBox[\(d\), \(_\)]\) -
     
116. \!\(\*OverscriptBox[\(o1\), \(_\)]\)) + o1];
     
117. Print["s=", s, ",    \!\(\*OverscriptBox[\(s\), \(_\)]\)=",
     
118. \!\(\*OverscriptBox[\(s\), \(_\)]\)];
     
119. Print["s1=", s1];
     
120. Print["s2=", s2];
     
121. Print["s1-s2=", Simplify[s1 - s2]];
     
122. Print["。。。。。。。。。。。。。。。。。。。。。。。。。。"];
     
123. Simplify[{1, o1,
     
124. \!\(\*OverscriptBox[\(o1\), \(_\)]\), , d,
     
125. \!\(\*OverscriptBox[\(\(d\)\(\ \)\), \(_\)]\), e,
     
126. \!\(\*OverscriptBox[\(\(e\)\(\ \)\), \(_\)]\), , x,
     
127. \!\(\*OverscriptBox[\(x\), \(_\)]\)}]
     
128. Simplify[{2, p,
     
129. \!\(\*OverscriptBox[\(p\), \(_\)]\), , q,
     
130. \!\(\*OverscriptBox[\(\(q\)\(\ \)\), \(_\)]\), , s,
     
131. \!\(\*OverscriptBox[\(s\), \(_\)]\)}]
     
132. Simplify[{3, k[s, e], -((s - o1)/(
     
133. \!\(\*OverscriptBox[\(e\), \(_\)]\) -
     
134. \!\(\*OverscriptBox[\(o1\), \(_\)]\))), , k[s, d],
     
135.   k[p, d], -((s - o1)/(
     
136. \!\(\*OverscriptBox[\(d\), \(_\)]\) -
     
137. \!\(\*OverscriptBox[\(o1\), \(_\)]\))), -((d - o1)/(
     
138. \!\(\*OverscriptBox[\(s\), \(_\)]\) -
     
139. \!\(\*OverscriptBox[\(o1\), \(_\)]\))), , -((s - o1)/(
     
140. \!\(\*OverscriptBox[\(d\), \(_\)]\) -
     
141. \!\(\*OverscriptBox[\(o1\), \(_\)]\))) + (d - o1)/(
     
142. \!\(\*OverscriptBox[\(s\), \(_\)]\) -
     
143. \!\(\*OverscriptBox[\(o1\), \(_\)]\))}](*验证S在圆O1上*)
     
144. Simplify[{4, (s - o1) (
     
145. \!\(\*OverscriptBox[\(s\), \(_\)]\) -
     
146. \!\(\*OverscriptBox[\(o1\), \(_\)]\)), , (x - o1) (
     
147. \!\(\*OverscriptBox[\(x\), \(_\)]\) -
     
148. \!\(\*OverscriptBox[\(o1\), \(_\)]\)), (o1 - a) (
     
149. \!\(\*OverscriptBox[\(o1\), \(_\)]\) -
     
150. \!\(\*OverscriptBox[\(a\), \(_\)]\)), , (e - o1) (
     
151. \!\(\*OverscriptBox[\(e\), \(_\)]\) -
     
152. \!\(\*OverscriptBox[\(o1\), \(_\)]\)), (d - o1) (
     
153. \!\(\*OverscriptBox[\(d\), \(_\)]\) -
     
154. \!\(\*OverscriptBox[\(o1\), \(_\)]\)), x
     
155. \!\(\*OverscriptBox[\(x\), \(_\)]\)}]
     
156. Simplify[{5, (s - o1) (
     
157. \!\(\*OverscriptBox[\(s\), \(_\)]\) -
     
158. \!\(\*OverscriptBox[\(o1\), \(_\)]\)) - (x - o1) (
     
159. \!\(\*OverscriptBox[\(x\), \(_\)]\) -
     
160. \!\(\*OverscriptBox[\(o1\), \(_\)]\))}]
     
161. Simplify[{6, k[a, d], k[a, e], -((a - o1)/(
     
162. \!\(\*OverscriptBox[\(d\), \(_\)]\) -
     
163. \!\(\*OverscriptBox[\(o1\), \(_\)]\)))}]
     
164. Simplify[{7, s1, s2, , s1 - s2}](*验证S在圆O1上*)
     
165. 
     

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部分运行结果如下：

dlsh

谢谢老朋友，找了很久没有发现，的确是那句错了，也可以改成 o1=Simplify[Jd[-b c , m , k[a, t], a ]];你的计算结论还说明S是△APQ的垂心，其它结果可能还可以发现新的几何关系，奇怪为什么原来的错误程序P和Q点的表达式却是正确的？是否可能不利用T点来证明？写了一封信给你，你注意查收。

dlsh

下面是用向量商的结果

1. 
   
2. Clear["Global`*"](*v=-1/a^2;*)
   
3. b = a^2 v^2;(*这句是根据AD=AE判定O1在角DAE平分线上，由Factor[k[o1,a]^2-a^2b c]可以推出*)
   
4. 
   
5. \!\(\*OverscriptBox["o", "_"]\) = o = 0;
   
6. \!\(\*OverscriptBox["c", "_"]\) = c = 1;
   
7. \!\(\*OverscriptBox["a", "_"]\) = 1/a; b = a^2 v^2;
   
8. \!\(\*OverscriptBox["b", "_"]\) = 1/b;
   
9. \!\(\*OverscriptBox["v", "_"]\) = 1/v; h = a + b + c;
   
10. \!\(\*OverscriptBox["h", "_"]\) =
    
11. \!\(\*OverscriptBox["a", "_"]\) +
    
12. \!\(\*OverscriptBox["b", "_"]\) +
    
13. \!\(\*OverscriptBox["c", "_"]\); m = (a + h)/2;
    
14. \!\(\*OverscriptBox["m", "_"]\) = (
    
15. \!\(\*OverscriptBox["a", "_"]\) +
    
16. \!\(\*OverscriptBox["h", "_"]\))/2;
    
17. o1 = (h - a v)/(1 - v);
    
18. \!\(\*OverscriptBox["o1", "_"]\) = (
    
19. \!\(\*OverscriptBox["h", "_"]\) -
    
20. \!\(\*OverscriptBox["a", "_"]\)
    
21. \!\(\*OverscriptBox["v", "_"]\))/(1 -
    
22. \!\(\*OverscriptBox["v", "_"]\));(*假设
    
23. \!\(\*OverscriptBox["O1H", "\[RightVector]"]\)/
    
24. \!\(\*OverscriptBox["O1A", "\[RightVector]"]\)=v*)
    
25. k[a_, b_] := (a - b)/(
    
26. \!\(\*OverscriptBox["a", "_"]\) -
    
27. \!\(\*OverscriptBox["b", "_"]\));
    
28. \!\(\*OverscriptBox["k", "_"]\)[a_, b_] := 1/k[a, b];(*复斜率定义*)
    
29. 
    
30. \!\(\*OverscriptBox["Jd", "_"]\)[k1_, a1_, k2_, a2_] := -((a1 - k1
    
31. \!\(\*OverscriptBox["a1", "_"]\) - (a2 - k2
    
32. \!\(\*OverscriptBox["a2", "_"]\)))/(k1 - k2));
    
33. (*复斜率等于k1,过点A1与复斜率等于k2,过点A2的直线交点*)
    
34. Jd[k1_, a1_, k2_, a2_] := -((k2 (a1 - k1
    
35. \!\(\*OverscriptBox["a1", "_"]\)) - k1 (a2 - k2
    
36. \!\(\*OverscriptBox["a2", "_"]\)))/(k1 - k2));
    
37. FourPoint[a_, b_, c_, d_] := ((
    
38. \!\(\*OverscriptBox["c", "_"]\) d - c
    
39. \!\(\*OverscriptBox["d", "_"]\)) (a - b) - (
    
40. \!\(\*OverscriptBox["a", "_"]\) b - a
    
41. \!\(\*OverscriptBox["b", "_"]\)) (c - d))/((a - b) (
    
42. \!\(\*OverscriptBox["c", "_"]\) -
    
43. \!\(\*OverscriptBox["d", "_"]\)) - (
    
44. \!\(\*OverscriptBox["a", "_"]\) -
    
45. \!\(\*OverscriptBox["b", "_"]\)) (c - d));(*过两点A和B、C和D的交点*)
    
46. 
    
47. \!\(\*OverscriptBox["FourPoint", "_"]\)[a_, b_, c_, d_] := -(((c
    
48. \!\(\*OverscriptBox["d", "_"]\) -
    
49. \!\(\*OverscriptBox["c", "_"]\) d) (
    
50. \!\(\*OverscriptBox["a", "_"]\) -
    
51. \!\(\*OverscriptBox["b", "_"]\)) - ( a
    
52. \!\(\*OverscriptBox["b", "_"]\) -
    
53. \!\(\*OverscriptBox["a", "_"]\) b) (
    
54. \!\(\*OverscriptBox["c", "_"]\) -
    
55. \!\(\*OverscriptBox["d", "_"]\)))/((a - b) (
    
56. \!\(\*OverscriptBox["c", "_"]\) -
    
57. \!\(\*OverscriptBox["d", "_"]\)) - (
    
58. \!\(\*OverscriptBox["a", "_"]\) -
    
59. \!\(\*OverscriptBox["b", "_"]\)) (c - d)));
    
60. Duichengdian[a_, b_, p_] := (
    
61. \!\(\*OverscriptBox["a", "_"]\) b - a
    
62. \!\(\*OverscriptBox["b", "_"]\) +
    
63. \!\(\*OverscriptBox["p", "_"]\) (a - b))/(
    
64. \!\(\*OverscriptBox["a", "_"]\) -
    
65. \!\(\*OverscriptBox["b", "_"]\));(*P关于AB的对称点*)
    
66. 
    
67. \!\(\*OverscriptBox["Duichengdian", "_"]\)[a_, b_, p_] := (a
    
68. \!\(\*OverscriptBox["b", "_"]\) -
    
69. \!\(\*OverscriptBox["a", "_"]\) b + p (
    
70. \!\(\*OverscriptBox["a", "_"]\) -
    
71. \!\(\*OverscriptBox["b", "_"]\)))/(a - b);
    
72.    (* o1=Jd[ - b c,m,-a t,a];
    
73. \!\(\*OverscriptBox["o1", "_"]\)=
    
74. \!\(\*OverscriptBox["Jd", "_"]\)[- b c,m,-a t,a];o1=Jd[a b,m,-a t,a];
    
75. \!\(\*OverscriptBox["o1", "_"]\)=
    
76. \!\(\*OverscriptBox["Jd", "_"]\)[a b,m,-a t,a];*)
    
77. d = a b (
    
78. \!\(\*OverscriptBox["a", "_"]\) -
    
79. \!\(\*OverscriptBox["o1", "_"]\)) + o1;
    
80. \!\(\*OverscriptBox["d", "_"]\) =
    
81. \!\(\*OverscriptBox["a", "_"]\)
    
82. \!\(\*OverscriptBox["b", "_"]\) (a - o1) +
    
83. \!\(\*OverscriptBox["o1", "_"]\); e = a c (
    
84. \!\(\*OverscriptBox["a", "_"]\) -
    
85. \!\(\*OverscriptBox["o1", "_"]\)) + o1;
    
86. \!\(\*OverscriptBox["e", "_"]\) =
    
87. \!\(\*OverscriptBox["a", "_"]\)
    
88. \!\(\*OverscriptBox["c", "_"]\) (a - o1) +
    
89. \!\(\*OverscriptBox["o1", "_"]\);
    
90. x = Duichengdian[o, o1, a];
    
91. \!\(\*OverscriptBox["x", "_"]\) =
    
92. \!\(\*OverscriptBox["Duichengdian", "_"]\)[o, o1, a];
    
93. q = -(k[x, d]/x);
    
94. \!\(\*OverscriptBox["q", "_"]\) = 1/q; p = -(k[x, e]/x);
    
95. \!\(\*OverscriptBox["p", "_"]\) = 1/p;
    
96. s = FourPoint[q, e, p, d];
    
97. \!\(\*OverscriptBox["s", "_"]\) =
    
98. \!\(\*OverscriptBox["FourPoint", "_"]\)[q, e, p, d];
    
99. s1 = -k[q, e] (
    
100. \!\(\*OverscriptBox["e", "_"]\) -
     
101. \!\(\*OverscriptBox["o1", "_"]\)) + o1; s2 = -k[p, d] (
     
102. \!\(\*OverscriptBox["d", "_"]\) -
     
103. \!\(\*OverscriptBox["o1", "_"]\)) + o1;
     
104. Simplify[{1, o1,
     
105. \!\(\*OverscriptBox["o1", "_"]\), , d,
     
106. \!\(\*OverscriptBox[
     
107. RowBox[{"d", "\[IndentingNewLine]"}], "_"]\), e,
     
108. \!\(\*OverscriptBox[
     
109. RowBox[{"e", "\[IndentingNewLine]"}], "_"]\), , x,
     
110. \!\(\*OverscriptBox["x", "_"]\)}]
     
111. Simplify[{2, p,
     
112. \!\(\*OverscriptBox["p", "_"]\), , q,
     
113. \!\(\*OverscriptBox[
     
114. RowBox[{"q", "\[IndentingNewLine]"}], "_"]\), , s,
     
115. \!\(\*OverscriptBox["s", "_"]\)}]
     
116. Simplify[{3, k[s, e], -((s - o1)/(
     
117. \!\(\*OverscriptBox["e", "_"]\) -
     
118. \!\(\*OverscriptBox["o1", "_"]\))), , k[s, d], k[p, d], -((s - o1)/(
     
119. \!\(\*OverscriptBox["d", "_"]\) -
     
120. \!\(\*OverscriptBox["o1", "_"]\))), -((d - o1)/(
     
121. \!\(\*OverscriptBox["s", "_"]\) -
     
122. \!\(\*OverscriptBox["o1", "_"]\))), , -((s - o1)/(
     
123. \!\(\*OverscriptBox["d", "_"]\) -
     
124. \!\(\*OverscriptBox["o1", "_"]\))) + (d - o1)/(
     
125. \!\(\*OverscriptBox["s", "_"]\) -
     
126. \!\(\*OverscriptBox["o1", "_"]\))}](*验证S在圆O1上*)
     
127. Simplify[{4, (s - o1) (
     
128. \!\(\*OverscriptBox["s", "_"]\) -
     
129. \!\(\*OverscriptBox["o1", "_"]\)), (x - o1) (
     
130. \!\(\*OverscriptBox["x", "_"]\) -
     
131. \!\(\*OverscriptBox["o1", "_"]\)), (o1 - a) (
     
132. \!\(\*OverscriptBox["o1", "_"]\) -
     
133. \!\(\*OverscriptBox["a", "_"]\)), , (e - o1) (
     
134. \!\(\*OverscriptBox["e", "_"]\) -
     
135. \!\(\*OverscriptBox["o1", "_"]\)), (d - o1) (
     
136. \!\(\*OverscriptBox["d", "_"]\) -
     
137. \!\(\*OverscriptBox["o1", "_"]\)), x
     
138. \!\(\*OverscriptBox["x", "_"]\)}]
     
139. Simplify[{5, (s - o1) (
     
140. \!\(\*OverscriptBox["s", "_"]\) -
     
141. \!\(\*OverscriptBox["o1", "_"]\)) - (x - o1) (
     
142. \!\(\*OverscriptBox["x", "_"]\) -
     
143. \!\(\*OverscriptBox["o1", "_"]\))}]
     
144. Simplify[{6, k[a, d], k[a, e], -((a - o1)/(
     
145. \!\(\*OverscriptBox["d", "_"]\) -
     
146. \!\(\*OverscriptBox["o1", "_"]\)))}]
     
147. Simplify[{7, s1, s2}]
     
148. Simplify[k[o1, a]^2 - a^2 b c]
     
149. Factor[k[o1, a]^2 - a^2 b c]
     
150. 
     
151. 
     
152. 
     

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TSC999

收到了您发给我邮箱的邮件。还没有来得及看各个附件的内容。关于向量商、共轭比的说法，我也觉得不如称为复斜率更简单、清晰，容易被人接受。您以前那篇论文，内容太少了，完全能把内容扩充十倍以上，主要是列出用于机器证明的公式，构图的方法。
我对复斜率的理解见下图。