⊙I内切于ΔABC，D为AC中点，AE⊥BI，DJ∥AE，⊙DEF交DJ于D,K，证 ⊙JFK 与 ⊙I 相切

有好好生活吗

ΔABC 中，圆 I 是其内切圆，AC＞BC＞AB，D 为 AC 中点，M 为圆 I 与 AC 切点，

F 在 AM 上，AE⊥BI 于点 E，DJ∥AE 且 JEF 三点共线，圆 DEF 交直线 DJ 于 D、K 两点，

证明：圆 JFK 与圆 I 相切。

denglongshan

能否画图

denglongshan

线性构造问题，一步步构造即可

denglongshan

denglongshan 发表于 2021-8-15 21:20
能否画图

如果你会用软件 ，代数方法不难，因为是线性构造问题。

denglongshan

1. 
   
2. \!\(\*OverscriptBox["i", "_"]\) = i = 0;
   
3. \!\(\*OverscriptBox["m", "_"]\) = 1/m;
   
4. a = (2 p m)/(m + p);
   
5. \!\(\*OverscriptBox["a", "_"]\) = 2/(m + p); b = (2 p n)/(n + p);
   
6. \!\(\*OverscriptBox["b", "_"]\) = 2/(n + p); c = (2 n m)/(n + m);
   
7. \!\(\*OverscriptBox["c", "_"]\) = 2/(m + n);
   
8. d = (a + c)/2;
   
9. \!\(\*OverscriptBox["d", "_"]\) = (
   
10. \!\(\*OverscriptBox["a", "_"]\) +
    
11. \!\(\*OverscriptBox["c", "_"]\))/2; f = (a - \[Lambda] d)/(
    
12. 1 - \[Lambda]);
    
13. \!\(\*OverscriptBox["f", "_"]\) = (
    
14. \!\(\*OverscriptBox["a", "_"]\) - \[Lambda]
    
15. \!\(\*OverscriptBox["d", "_"]\))/(1 - \[Lambda]);(*假设
    
16. \!\(\*OverscriptBox["FA", "\[RightVector]"]\)：F
    
17. \!\(\*OverscriptBox["D", "\[RightVector]"]\)=\[Lambda]*)
    
18. 
    
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. 
    
24. \!\(\*OverscriptBox["Jd", "_"]\)[k1_, a1_, k2_, a2_] := -((a1 - k1
    
25. \!\(\*OverscriptBox["a1", "_"]\) - (a2 - k2
    
26. \!\(\*OverscriptBox["a2", "_"]\)))/(k1 - k2));
    
27. (*复斜率等于k1,过点A1与复斜率等于k2,过点A2的直线交点*)
    
28. Jd[k1_, a1_, k2_, a2_] := -((k2 (a1 - k1
    
29. \!\(\*OverscriptBox["a1", "_"]\)) - k1 (a2 - k2
    
30. \!\(\*OverscriptBox["a2", "_"]\)))/(k1 - k2));
    
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. (*过两点A和B、C和D的交点*)
    
41. 
    
42. \!\(\*OverscriptBox["FourPoint", "_"]\)[a_, b_, c_, d_] := -(((c
    
43. \!\(\*OverscriptBox["d", "_"]\) -
    
44. \!\(\*OverscriptBox["c", "_"]\) d) (
    
45. \!\(\*OverscriptBox["a", "_"]\) -
    
46. \!\(\*OverscriptBox["b", "_"]\)) - ( a
    
47. \!\(\*OverscriptBox["b", "_"]\) -
    
48. \!\(\*OverscriptBox["a", "_"]\) b) (
    
49. \!\(\*OverscriptBox["c", "_"]\) -
    
50. \!\(\*OverscriptBox["d", "_"]\)))/((a - b) (
    
51. \!\(\*OverscriptBox["c", "_"]\) -
    
52. \!\(\*OverscriptBox["d", "_"]\)) - (
    
53. \!\(\*OverscriptBox["a", "_"]\) -
    
54. \!\(\*OverscriptBox["b", "_"]\)) (c - d)));
    
55. XiangjiaoxuanLianxin[o1_, a_, o2_, b_] := 1/(2 (
    
56. \!\(\*OverscriptBox["o2", "_"]\) -
    
57. \!\(\*OverscriptBox["o1", "_"]\))) (a
    
58. \!\(\*OverscriptBox["a", "_"]\) - b
    
59. \!\(\*OverscriptBox["b", "_"]\) +
    
60. \!\(\*OverscriptBox["b", "_"]\) o2 + b
    
61. \!\(\*OverscriptBox["o2", "_"]\) -
    
62. \!\(\*OverscriptBox["a", "_"]\) o1 - a
    
63. \!\(\*OverscriptBox["o1", "_"]\) +
    
64. \!\(\*OverscriptBox["o2", "_"]\) o1 - o2
    
65. \!\(\*OverscriptBox["o1", "_"]\));(*圆 （O1，A）与圆 （O2，B)连心线与公共弦的交点*)
    
66. 
    
67. \!\(\*OverscriptBox["XiangjiaoxuanLianxin", "_"]\)[o1_, a_, o2_, b_] :=
    
68.    1/(2 (o2 - o1)) (a
    
69. \!\(\*OverscriptBox["a", "_"]\) - b
    
70. \!\(\*OverscriptBox["b", "_"]\) +
    
71. \!\(\*OverscriptBox["b", "_"]\) o2 + b
    
72. \!\(\*OverscriptBox["o2", "_"]\) -
    
73. \!\(\*OverscriptBox["a", "_"]\) o1 - a
    
74. \!\(\*OverscriptBox["o1", "_"]\) + o2
    
75. \!\(\*OverscriptBox["o1", "_"]\) -
    
76. \!\(\*OverscriptBox["o2", "_"]\) o1);
    
77. Chuizu[a_, b_, p_] := (
    
78. \!\(\*OverscriptBox["a", "_"]\) b - a
    
79. \!\(\*OverscriptBox["b", "_"]\) + p (
    
80. \!\(\*OverscriptBox["a", "_"]\) -
    
81. \!\(\*OverscriptBox["b", "_"]\)) +
    
82. \!\(\*OverscriptBox["p", "_"]\) (a - b))/(2 (
    
83. \!\(\*OverscriptBox["a", "_"]\) -
    
84. \!\(\*OverscriptBox["b", "_"]\)));(*=(1/2)[p+(
    
85. \!\(\*OverscriptBox["a", "_"]\)b-a
    
86. \!\(\*OverscriptBox["b", "_"]\)+
    
87. \!\(\*OverscriptBox["p", "_"]\)(a-b))/(
    
88. \!\(\*OverscriptBox["a", "_"]\)-
    
89. \!\(\*OverscriptBox["b", "_"]\))]P到直线AB的垂足*)
    
90. 
    
91. \!\(\*OverscriptBox["Chuizu", "_"]\)[a_, b_, p_] := (a
    
92. \!\(\*OverscriptBox["b", "_"]\) -
    
93. \!\(\*OverscriptBox["a", "_"]\) b +
    
94. \!\(\*OverscriptBox["p", "_"]\) (a - b) + p (
    
95. \!\(\*OverscriptBox["a", "_"]\) -
    
96. \!\(\*OverscriptBox["b", "_"]\)))/(2 (a - b));
    
97. (*Duichendian[a_,b_,p_]:=(
    
98. \!\(\*OverscriptBox["a", "_"]\)b-a
    
99. \!\(\*OverscriptBox["b", "_"]\)+
    
100. \!\(\*OverscriptBox["p", "_"]\)(a-b))/(
     
101. \!\(\*OverscriptBox["a", "_"]\)-
     
102. \!\(\*OverscriptBox["b", "_"]\));P关于直线AB的对称点
     
103. 
     
104. \!\(\*OverscriptBox["Duichendian", "_"]\)[a_,b_,p_]:=(
     
105. \!\(\*OverscriptBox["b", "_"]\)-
     
106. \!\(\*OverscriptBox["a", "_"]\)b+
     
107. \!\(\*OverscriptBox["p", "_"]\)(a-b))/(a-b);*)
     
108. Wx[a_, b_, c_] := (a (-b + c)
     
109. \!\(\*OverscriptBox["a", "_"]\) + b (a - c)
     
110. \!\(\*OverscriptBox["b", "_"]\) + (-a + b) c
     
111. \!\(\*OverscriptBox["c", "_"]\))/((-b + c)
     
112. \!\(\*OverscriptBox["a", "_"]\) + (a - c)
     
113. \!\(\*OverscriptBox["b", "_"]\) + (-a + b)
     
114. \!\(\*OverscriptBox["c", "_"]\));
     
115. \!\(\*OverscriptBox["Wx", "_"]\)[a_, b_, c_] := ((b - c)
     
116. \!\(\*OverscriptBox["b", "_"]\)
     
117. \!\(\*OverscriptBox["c", "_"]\) +
     
118. \!\(\*OverscriptBox["a", "_"]\) ((a - b)
     
119. \!\(\*OverscriptBox["b", "_"]\) + (-a + c)
     
120. \!\(\*OverscriptBox["c", "_"]\)))/((-b + c)
     
121. \!\(\*OverscriptBox["a", "_"]\) + (a - c)
     
122. \!\(\*OverscriptBox["b", "_"]\) + (-a + b)
     
123. \!\(\*OverscriptBox["c", "_"]\));
     
124. 
     
125. e = Chuizu[i, b, a];
     
126. \!\(\*OverscriptBox["e", "_"]\) =
     
127. \!\(\*OverscriptBox["Chuizu", "_"]\)[i, b, a];
     
128. j = Jd[k[e, f], e, -p n, d];
     
129. \!\(\*OverscriptBox["j", "_"]\) =
     
130. \!\(\*OverscriptBox["Jd", "_"]\)[k[e, f], e, -p n, d];
     
131. o = Wx[d, e, f];
     
132. \!\(\*OverscriptBox[
     
133. StyleBox["o",
     
134. FontSize->14], "_"]\) =
     
135. \!\(\*OverscriptBox["Wx", "_"]\)[d, e, f]; k = p  n (
     
136. \!\(\*OverscriptBox["d", "_"]\) -
     
137. \!\(\*OverscriptBox["o", "_"]\)) + o;
     
138. \!\(\*OverscriptBox["k", "_"]\) = (d - o)/(p n) +
     
139. \!\(\*OverscriptBox["o", "_"]\); q = Wx[j, k, f];
     
140. \!\(\*OverscriptBox[
     
141. StyleBox["q",
     
142. FontSize->24], "_"]\) =
     
143. \!\(\*OverscriptBox["Wx", "_"]\)[j, k, f];(*(d-o)/p n*)
     
144. t = XiangjiaoxuanLianxin[q, f, i, m];
     
145. \!\(\*OverscriptBox["t", "_"]\) =
     
146. \!\(\*OverscriptBox["XiangjiaoxuanLianxin", "_"]\)[q, f, i, m];
     
147. Simplify[{d,
     
148. \!\(\*OverscriptBox["d", "_"]\), e,
     
149. \!\(\*OverscriptBox["e", "_"]\), f,
     
150. \!\(\*OverscriptBox["f", "_"]\), 1, j,
     
151. \!\(\*OverscriptBox["j", "_"]\), o,
     
152. \!\(\*OverscriptBox["o", "_"]\), k,
     
153. \!\(\*OverscriptBox["k", "_"]\), 2, q,
     
154. \!\(\*OverscriptBox["q", "_"]\), 3, t,
     
155. \!\(\*OverscriptBox["t", "_"]\), , t  
     
156. \!\(\*OverscriptBox["t", "_"]\)}]
     
157. Factor[{d,
     
158. \!\(\*OverscriptBox["d", "_"]\), e,
     
159. \!\(\*OverscriptBox["e", "_"]\), f,
     
160. \!\(\*OverscriptBox["f", "_"]\), 1, j,
     
161. \!\(\*OverscriptBox["j", "_"]\), o,
     
162. \!\(\*OverscriptBox["o", "_"]\), k,
     
163. \!\(\*OverscriptBox["k", "_"]\), 2, q,
     
164. \!\(\*OverscriptBox["q", "_"]\), 3, t,
     
165. \!\(\*OverscriptBox["t", "_"]\), , t  
     
166. \!\(\*OverscriptBox["t", "_"]\)}]
     
167. 
     
168. 
     

复制代码

再运行一次就不对了，不知道原因，Mathematica7.0版本

denglongshan

以上证明说明AC＞BC＞AB不必要