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9点圆

9点圆
几何画板及绘图内容加载中，请稍等……
2016-05-29 11:51
测试改名0圆几何作图
这个范例演示了三角形的9点圆。9点圆是三角形的三个边的中点、三个高的垂足以及三个中线的中点共9个点所在的圆。核心数学概念包括三角形的中点、垂足、中线和圆。通过使用GeoGebra命令如Segment, Midpoint, Circle等，可以构造出9点圆。适合初中及以上学生学习几何作图。

GeoGebra 命令及参数

  1. Segment(A, B) → f
  2. Segment(B, C_1) → g
  3. Segment(C_1, A) → h
  4. Midpoint(f) → D
  5. Midpoint(h) → E
  6. Midpoint(g) → F
  7. Min(list1) → a
  8. Circle(D, (0.15 * a)) → c
  9. Circle(E, (0.15 * a)) → d
  10. Circle(F, (0.15 * a)) → e
  11. Intersect(c, f, 2) → H
  12. Intersect(d, h, 2) → K
  13. Intersect(e, g, 2) → M
  14. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], D) → A'
  15. Dilate(B, If[2 < b ≤ 3, 3 - b, 0], F) → B'
  16. Dilate(C_1, If[4 < b ≤ 5, 5 - b, 0], E) → C'
  17. Segment(A, A') → l
  18. Segment(B, B') → m
  19. Segment(C_1, C') → n
  20. Rotate(A, If[1 < b ≤ 2, ((-(b - 1)) * pi), (-pi)], D) → A'_1
  21. Rotate(A', If[1 < b ≤ 2, ((b - 1) * pi), pi], D) → A''
  22. Segment(A'_1, A'') → l'
  23. Rotate(B, If[3 < b ≤ 4, ((-(b - 3)) * pi), (-pi)], F) → B'_1
  24. Rotate(B', If[3 < b ≤ 4, ((b - 3) * pi), pi], F) → B''
  25. Segment(B'_1, B'') → m'
  26. Rotate(C_1, If[5 < b ≤ 6, ((-(b - 5)) * pi), (-pi)], E) → C'_1
  27. Rotate(C', If[5 < b ≤ 6, ((b - 5) * pi), pi], E) → C''
  28. Segment(C'_1, C'') → n'
  29. Polygon(A, B, C_1) → poly1, c_1, a_1, b_1
  30. Line(A, B) → i
  31. Point(i) → C
  32. Point(i) → G
  33. OrthogonalLine(C_1, i) → j
  34. Intersect(i, j) → I
  35. Ray(A, G) → k
  36. Ray(B, C) → p
  37. Point(k) → J
  38. Point(p) → L
  39. Ray(A, J) → q
  40. Ray(B, L) → r
  41. Circle(I, (0.15 * a)) → s
  42. Intersect(s, i, 2) → N
  43. Intersect(s, j, 2) → O
  44. Angle(C_1, B, A) → α
  45. Angle(B, A, C_1) → β
  46. Angle(A, C_1, B) → γ
  47. Line(A, C_1) → f_1
  48. Point(f_1) → P
  49. Point(f_1) → Q
  50. Ray(A, P) → g_1
  51. Ray(C_1, Q) → h_1
  52. Point(g_1) → R
  53. Ray(A, R) → i_1
  54. Point(h_1) → S
  55. Ray(C_1, S) → j_1
  56. OrthogonalLine(B, f_1) → k_1
  57. Intersect(f_1, k_1) → T
  58. Circle(T, (0.15 * a)) → d_1
  59. Intersect(d_1, f_1, 1) → U
  60. Intersect(d_1, k_1, 1) → V
  61. Line(C_1, B) → l_1
  62. OrthogonalLine(A, l_1) → m_1
  63. Intersect(l_1, m_1) → W
  64. Circle(W, (0.15 * a)) → p_1
  65. Intersect(p_1, l_1, 1) → Z
  66. Intersect(p_1, m_1, 1) → A_1
  67. Point(l_1) → B_1
  68. Point(l_1) → D_1
  69. Ray(C_1, B_1) → n_1
  70. Ray(B, D_1) → r_1
  71. Point(n_1) → E_1
  72. Point(r_1) → F_1
  73. Ray(C_1, E_1) → s_1
  74. Ray(B, F_1) → t_1
  75. Dilate(C_1, If[6 < b ≤ 7, 7 - b, 0], I) → G_1
  76. Segment(C_1, G_1) → f_2
  77. Dilate(B, If[7 < b ≤ 8, 8 - b, 0], T) → B'_2
  78. Segment(B, B'_2) → g_2
  79. Dilate(A, If[8 < b ≤ 9, 9 - b, 0], W) → A'_2
  80. Segment(A, A'_2) → h_2
  81. Intersect(k_1, m_1) → H_1
  82. Dilate(I, If[12 < b ≤ 13, 13 - b, 0], H_1) → I'
  83. Dilate(A, If[12 < b ≤ 13, 13 - b, 0], H_1) → A'_3
  84. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], H_1) → T'
  85. Segment(I, I') → i_2
  86. Segment(A, A'_3) → j_2
  87. Segment(T, T') → k_2
  88. Dilate(C_1, If[12 < b ≤ 13, 13 - b, 0], H_1) → I_1
  89. Dilate(W, If[12 < b ≤ 13, 13 - b, 0], H_1) → W'
  90. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], H_1) → T'_1
  91. Segment(W, W') → l_2
  92. Segment(C_1, I_1) → m_2
  93. Segment(T, T'_1) → n_2
  94. Dilate(I, If[12 < b ≤ 13, 13 - b, 0], H_1) → I'_1
  95. Dilate(B, If[12 < b ≤ 13, 13 - b, 0], H_1) → B'_3
  96. Dilate(W, If[12 < b ≤ 13, 13 - b, 0], H_1) → W'_1
  97. Segment(B, B'_3) → p_2
  98. Segment(W, W'_1) → q_2
  99. Segment(I, I'_1) → r_2
  100. Midpoint(H_1, C_1) → J_1
  101. Midpoint(H_1, A) → K_1
  102. Midpoint(H_1, B) → L_1
  103. Dilate(H_1, If[14 < b ≤ 15, 15 - b, 0], J_1) → M_1
  104. Segment(H_1, M_1) → s_2
  105. Dilate(J_1, If[15 < b ≤ 16, 16 - b, 0], C_1) → N_1
  106. Segment(J_1, N_1) → t_2
  107. Rotate(H_1, If[16 < b ≤ 17, ((b - 16) * pi), pi], J_1) → O_1
  108. Rotate(M_1, If[16 < b ≤ 17, ((b - 16) * pi), pi], J_1) → P_1
  109. Segment(O_1, P_1) → a_2
  110. Dilate(H_1, If[17 < b ≤ 18, 18 - b, 0], L_1) → Q_1
  111. Segment(H_1, Q_1) → b_2
  112. Dilate(L_1, If[18 < b ≤ 19, 19 - b, 0], B) → R_1
  113. Segment(L_1, R_1) → e_2
  114. Rotate(H_1, If[19 < b ≤ 20, ((b - 19) * pi), pi], L_1) → S_1
  115. Rotate(Q_1, If[19 < b ≤ 20, ((b - 19) * pi), pi], L_1) → T_1
  116. Segment(S_1, T_1) → f_3
  117. Dilate(H_1, If[20 < b ≤ 21, 21 - b, 0], K_1) → U_1
  118. Segment(H_1, U_1) → g_3
  119. Dilate(K_1, If[21 < b ≤ 22, 22 - b, 0], A) → V_1
  120. Segment(K_1, V_1) → h_3
  121. Rotate(H_1, If[22 < b ≤ 23, ((b - 22) * pi), pi], K_1) → W_1
  122. Rotate(U_1, If[22 < b ≤ 23, ((b - 22) * pi), pi], K_1) → Z_1
  123. Segment(W_1, Z_1) → i_3
  124. Intersect(e, l_1) → A_2, B_2
  125. OrthogonalLine(F, l_1) → j_3
  126. Intersect(e, j_3) → C_2, D_2
  127. OrthogonalLine(E, f_1) → k_3
  128. Circle(E, (0.15 * a)) → d_3
  129. Intersect(d_3, k_3) → E_2, F_2
  130. Intersect(d_3, f_1) → G_2, H_2
  131. Intersect(k_3, j_3) → I_2
  132. Midpoint(H_1, I_2) → J_2
  133. Dilate(H_1, If[28 < b ≤ 29, 29 - b, 0], J_2) → K_2
  134. Segment(H_1, K_2) → l_3
  135. Dilate(J_2, If[29 < b ≤ 30, 30 - b, 0], I_2) → L_2
  136. Segment(J_2, L_2) → m_3
  137. Rotate(H_1, If[30 < b ≤ 31, ((b - 30) * pi), pi], J_2) → M_2
  138. Rotate(K_2, If[30 < b ≤ 31, ((b - 30) * pi), pi], J_2) → N_2
  139. Segment(M_2, N_2) → n_3
  140. Dilate(J_2, If[31 < b ≤ 32, 32 - b, 0], D) → O_2
  141. Segment(J_2, O_2) → r_3
  142. Rotate(J_2, If[32 < b ≤ 34, ((b - 32) * pi), (2 * pi)], J_2) → P_2
  143. Rotate(O_2, If[32 < b ≤ 34, ((b - 32) * pi), (2 * pi)], J_2) → Q_2
  144. Segment(P_2, Q_2) → s_3
  145. CircleArc(J_2, D, Q_2) → t_3
  146. Circle(J_2, D) → c_4
  147. Segment(B, T) → a_3
  148. Segment(I, C_1) → b_3
  149. Segment(A, W) → f_4
  150. Segment(H_1, I) → g_4
  151. Segment(H_1, T) → h_4
  152. Segment(H_1, W) → i_4
  153. Angle(B, T, C_1) → δ
  154. Angle(B, I, C_1) → ε
  155. Angle(A, W, B) → ζ
  156. Angle(I_2, E, C_1) → η
  157. Angle(I_2, F, B) → θ