积极 AI Logo 积极 AI

九点圆构造与性质演示

九点圆构造与性质演示
几何画板及绘图内容加载中，请稍等……
2016-09-14 09:33
测试改名0圆解析几何几何作图
①本范例演示了如何通过构造三角形的三条高线交点、各边中点及各顶点连接线段的中点，绘制九点圆，并展示其与三角形外接圆的关系；②涉及核心数学概念包括三角形、垂心、中点、圆、几何作图、极限等；③作图要点提炼自GeoGebra命令：Segment（线段）、Midpoint（中点）、Circle（圆）、Intersect（交点）、Dilate（缩放）、Rotate（旋转）、Polygon（多边形）、Line（直线）、Point（点）、OrthogonalLine（垂线）、Ray（射线）、CircleSector（扇形）、Angle（角度）、CircleArc（圆弧）；④适合高中及以上数学学习者，用于理解九点圆的构造方法及其在几何证明中的应用。

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, (0.1 * a)) → c
  9. Circle(E, (0.1 * a)) → d
  10. Circle(F, (0.1 * 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) * π), π], D) → A'_1
  21. Rotate(A', If[1 < b ≤ 2, ((b - 1) * π), π], D) → A''
  22. Segment(A'_1, A'') → l'
  23. Rotate(B, If[3 < b ≤ 4, ((b - 3) * π), π], F) → B'_1
  24. Rotate(B', If[3 < b ≤ 4, ((b - 3) * π), π], F) → B''
  25. Segment(B'_1, B'') → m'
  26. Rotate(C_1, If[5 < b ≤ 6, ((b - 5) * π), π], E) → C'_1
  27. Rotate(C', If[5 < b ≤ 6, ((b - 5) * π), π], 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, (0.1 * a)) → s
  42. Intersect(s, i, 2) → N
  43. Intersect(s, j, 2) → O
  44. CircleSector(I, N, O) → t
  45. Angle(C_1, B, A) → α
  46. Angle(B, A, C_1) → β
  47. Angle(A, C_1, B) → γ
  48. Line(A, C_1) → f_1
  49. Point(f_1) → P
  50. Point(f_1) → Q
  51. Ray(A, P) → g_1
  52. Ray(C_1, Q) → h_1
  53. Point(g_1) → R
  54. Ray(A, R) → i_1
  55. Point(h_1) → S
  56. Ray(C_1, S) → j_1
  57. OrthogonalLine(B, f_1) → k_1
  58. Intersect(f_1, k_1) → T
  59. Circle(T, (0.1 * a)) → d_1
  60. Intersect(d_1, f_1, 1) → U
  61. Intersect(d_1, k_1, 1) → V
  62. CircleSector(T, U, V) → e_1
  63. Line(C_1, B) → l_1
  64. OrthogonalLine(A, l_1) → m_1
  65. Intersect(l_1, m_1) → W
  66. Circle(W, (0.1 * a)) → p_1
  67. Intersect(p_1, l_1, 1) → Z
  68. Intersect(p_1, m_1, 1) → A_1
  69. Point(l_1) → B_1
  70. Point(l_1) → D_1
  71. CircleSector(W, Z, A_1) → q_1
  72. Ray(C_1, B_1) → n_1
  73. Ray(B, D_1) → r_1
  74. Point(n_1) → E_1
  75. Point(r_1) → F_1
  76. Ray(C_1, E_1) → s_1
  77. Ray(B, F_1) → t_1
  78. Dilate(C_1, If[6 < b ≤ 7, 7 - b, 0], I) → G_1
  79. Segment(C_1, G_1) → f_2
  80. Dilate(B, If[7 < b ≤ 8, 8 - b, 0], T) → B'_2
  81. Segment(B, B'_2) → g_2
  82. Dilate(A, If[8 < b ≤ 9, 9 - b, 0], W) → A'_2
  83. Segment(A, A'_2) → h_2
  84. Intersect(k_1, m_1) → H_1
  85. Rotate(t, If[9 < b ≤ 10, ((b - 9) * π / 2), π / 2], I) → t'
  86. Rotate(e_1, If[10 < b ≤ 11, ((b - 10) * π / 2), π / 2], T) → c_2
  87. Rotate(q_1, If[11 < b ≤ 12, ((b - 11) * π / 2), π / 2], W) → d_2
  88. Dilate(I, If[12 < b ≤ 13, 13 - b, 0], H_1) → I'
  89. Dilate(A, If[12 < b ≤ 13, 13 - b, 0], H_1) → A'_3
  90. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], H_1) → T'
  91. Segment(I, I') → i_2
  92. Segment(A, A'_3) → j_2
  93. Segment(T, T') → k_2
  94. Dilate(C_1, If[12 < b ≤ 13, 13 - b, 0], H_1) → I_1
  95. Dilate(W, If[12 < b ≤ 13, 13 - b, 0], H_1) → W'
  96. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], H_1) → T'_1
  97. Segment(W, W') → l_2
  98. Segment(C_1, I_1) → m_2
  99. Segment(T, T'_1) → n_2
  100. Dilate(I, If[12 < b ≤ 13, 13 - b, 0], H_1) → I'_1
  101. Dilate(B, If[12 < b ≤ 13, 13 - b, 0], H_1) → B'_3
  102. Dilate(W, If[12 < b ≤ 13, 13 - b, 0], H_1) → W'_1
  103. Segment(B, B'_3) → p_2
  104. Segment(W, W'_1) → q_2
  105. Segment(I, I'_1) → r_2
  106. Midpoint(H_1, C_1) → J_1
  107. Midpoint(H_1, A) → K_1
  108. Midpoint(H_1, B) → L_1
  109. Dilate(H_1, If[14 < b ≤ 15, 15 - b, 0], J_1) → M_1
  110. Segment(H_1, M_1) → s_2
  111. Dilate(J_1, If[15 < b ≤ 16, 16 - b, 0], C_1) → N_1
  112. Segment(J_1, N_1) → t_2
  113. Rotate(H_1, If[16 < b ≤ 17, ((b - 16) * π), π], J_1) → O_1
  114. Rotate(M_1, If[16 < b ≤ 17, ((b - 16) * π), π], J_1) → P_1
  115. Segment(O_1, P_1) → a_2
  116. Dilate(H_1, If[17 < b ≤ 18, 18 - b, 0], L_1) → Q_1
  117. Segment(H_1, Q_1) → b_2
  118. Dilate(L_1, If[18 < b ≤ 19, 19 - b, 0], B) → R_1
  119. Segment(L_1, R_1) → e_2
  120. Rotate(H_1, If[19 < b ≤ 20, ((b - 19) * π), π], L_1) → S_1
  121. Rotate(Q_1, If[19 < b ≤ 20, ((b - 19) * π), π], L_1) → T_1
  122. Segment(S_1, T_1) → f_3
  123. Dilate(H_1, If[20 < b ≤ 21, 21 - b, 0], K_1) → U_1
  124. Segment(H_1, U_1) → g_3
  125. Dilate(K_1, If[21 < b ≤ 22, 22 - b, 0], A) → V_1
  126. Segment(K_1, V_1) → h_3
  127. Rotate(H_1, If[22 < b ≤ 23, ((b - 22) * π), π], K_1) → W_1
  128. Rotate(U_1, If[22 < b ≤ 23, ((b - 22) * π), π], K_1) → Z_1
  129. Segment(W_1, Z_1) → i_3
  130. Intersect(e, l_1) → A_2, B_2
  131. OrthogonalLine(F, l_1) → j_3
  132. Intersect(e, j_3) → C_2, D_2
  133. CircleSector(F, A_2, C_2) → c_3
  134. OrthogonalLine(E, f_1) → k_3
  135. Circle(E, (0.1 * a)) → d_3
  136. Intersect(d_3, k_3) → E_2, F_2
  137. Intersect(d_3, f_1) → G_2, H_2
  138. CircleSector(E, G_2, E_2) → e_3
  139. Rotate(c_3, If[25 < b ≤ 26, ((b - 25) * π / 2), π / 2], F) → p_3
  140. Rotate(e_3, If[26 < b ≤ 27, ((b - 26) * π / 2), π / 2], E) → q_3
  141. Intersect(k_3, j_3) → I_2
  142. Midpoint(H_1, I_2) → J_2
  143. Dilate(H_1, If[28 < b ≤ 29, 29 - b, 0], J_2) → K_2
  144. Segment(H_1, K_2) → l_3
  145. Dilate(J_2, If[29 < b ≤ 30, 30 - b, 0], I_2) → L_2
  146. Segment(J_2, L_2) → m_3
  147. Rotate(H_1, If[30 < b ≤ 31, ((b - 30) * π), π], J_2) → M_2
  148. Rotate(K_2, If[30 < b ≤ 31, ((b - 30) * π), π], J_2) → N_2
  149. Segment(M_2, N_2) → n_3
  150. Dilate(J_2, If[31 < b ≤ 32, 32 - b, 0], D) → O_2
  151. Segment(J_2, O_2) → r_3
  152. Rotate(J_2, If[32 < b ≤ 34, ((b - 32) * π), (2 * π)], J_2) → P_2
  153. Rotate(O_2, If[32 < b ≤ 34, ((b - 32) * π), (2 * π)], J_2) → Q_2
  154. Segment(P_2, Q_2) → s_3
  155. CircleArc(J_2, D, Q_2) → t_3
  156. Circle(J_2, D) → c_4
  157. Segment(B, T) → a_3
  158. Segment(I, C_1) → b_3
  159. Segment(A, W) → f_4
  160. Segment(H_1, I) → g_4
  161. Segment(H_1, T) → h_4
  162. Segment(H_1, W) → i_4
  163. Polygon(L_1, D, E, J_1) → poly2, l_4, d_4, e_4, j_4
  164. Angle(J_1, L_1, D) → δ
  165. Angle(L_1, D, E) → ε
  166. Angle(D, E, J_1) → ζ
  167. Angle(E, J_1, L_1) → η
  168. Dilate(D, If[37 < b ≤ 38, 38 - b, 0], L_1) → D'
  169. Segment(D, D') → k_4
  170. Dilate(L_1, If[38 < b ≤ 39, 39 - b, 0], J_1) → R_2
  171. Segment(L_1, R_2) → m_4
  172. Dilate(J_1, If[39 < b ≤ 40, 40 - b, 0], E) → S_2
  173. Segment(J_1, S_2) → n_4
  174. Dilate(E, If[40 < b ≤ 41, 41 - b, 0], D) → E'
  175. Segment(E, E') → p_4
  176. Polygon(D, F, J_1, K_1) → poly3, d_5, f_5, j_5, k_5
  177. Angle(J_1, F, D) → θ
  178. Angle(F, D, K_1) → ι
  179. Angle(D, K_1, J_1) → κ
  180. Angle(K_1, J_1, F) → λ
  181. Dilate(F, If[42 < b ≤ 43, 43 - b, 0], D) → F'
  182. Segment(F, F') → q_4
  183. Dilate(D, If[43 < b ≤ 44, 44 - b, 0], K_1) → D'_1
  184. Segment(D, D'_1) → r_4
  185. Dilate(K_1, If[44 < b ≤ 45, 45 - b, 0], J_1) → T_2
  186. Segment(K_1, T_2) → s_4
  187. Dilate(J_1, If[45 < b ≤ 46, 46 - b, 0], F) → U_2
  188. Segment(J_1, U_2) → t_4
  189. Polygon(E, F, L_1, K_1) → poly4, e_5, f_6, l_5, k_6
  190. Dilate(L_1, If[47 < b ≤ 48, 48 - b, 0], F) → V_2
  191. Segment(L_1, V_2) → a_4
  192. Dilate(F, If[48 < b ≤ 49, 49 - b, 0], E) → F'_1
  193. Segment(F, F'_1) → b_4
  194. Dilate(E, If[49 < b ≤ 50, 50 - b, 0], K_1) → E'_1
  195. Segment(E, E'_1) → g_5
  196. Dilate(K_1, If[50 < b ≤ 51, 51 - b, 0], L_1) → W_2
  197. Segment(K_1, W_2) → h_5
  198. Angle(E, F, L_1) → μ
  199. Angle(F, L_1, K_1) → ν
  200. Angle(L_1, K_1, E) → ξ
  201. Angle(K_1, E, F) → ο

AI 绘图提示语参考

画一个三角形，标出其三条边的中点、三条高的中点以及顶点到垂心的中点，并画出这些点构成的九点圆。同时，展示九点圆与三角形外接圆的关系，并标注垂心、中点、圆等关键几何元素，按构造顺序依次绘制。
以上提示语为根据原始绘图命令整理而来，仅供参考。