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探究共点性

探究共点性
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2020-04-01 20:51
测试改名0几何作图
本范例通过构造三角形的不同类型（等腰三角形、等边三角形、直角三角形、不等边三角形）来探究共点性。核心数学概念包括重心、垂心、内心和外心。通过使用GeoGebra命令如Circle（圆）、Segment（线段）、Intersect（交点）、Point（点）、OrthogonalLine（垂线）、Midpoint（中点）、AngularBisector（角平分线）等，可以动态地观察这些三角形的特殊点是否共线。适合高中几何学习者和教师使用。

GeoGebra 命令及参数

  1. Circle(A, B) → c
  2. Segment(A, B) → f
  3. Circle(B, f) → d
  4. Intersect(c, d) → C, D, ,
  5. Segment(B, D) → g
  6. Segment(A, D) → h
  7. Segment(E, F) → i
  8. Point(i) → G
  9. Circle(E, G) → e
  10. Circle(F, Segment[G, E]) → k
  11. Intersect(e, k) → H, I, ,
  12. Segment(E, H) → j
  13. Segment(F, H) → l
  14. Segment(J, K) → m
  15. OrthogonalLine(J, m) → n
  16. Point(n) → L
  17. Segment(L, K) → p
  18. Segment(J, L) → q
  19. Segment(M, N) → r
  20. Segment(N, O) → s
  21. Segment(M, O) → t
  22. Midpoint(E, H) → P
  23. Midpoint(H, F) → Q
  24. Segment(E, Q) → a
  25. Segment(F, P) → b
  26. Midpoint(E, F) → R
  27. Segment(H, R) → f_1
  28. Midpoint(B, D) → S
  29. Midpoint(D, A) → T
  30. Midpoint(B, A) → U
  31. Segment(A, S) → g_1
  32. Segment(B, T) → h_1
  33. Segment(U, D) → i_1
  34. Midpoint(J, K) → V
  35. Midpoint(L, J) → W
  36. Midpoint(L, K) → Z
  37. Segment(J, Z) → j_1
  38. Segment(L, V) → k_1
  39. Segment(K, W) → l_1
  40. Midpoint(M, O) → A_1
  41. Midpoint(M, N) → B_1
  42. Midpoint(N, O) → C_1
  43. Segment(M, C_1) → m_1
  44. Segment(N, A_1) → n_1
  45. Segment(O, B_1) → p_1
  46. OrthogonalLine(P, j) → q_1
  47. OrthogonalLine(Q, l) → r_1
  48. OrthogonalLine(R, i) → s_1
  49. Intersect(q_1, i) → D_1
  50. Segment(P, D_1) → t_1
  51. Intersect(r_1, i) → E_1
  52. Segment(Q, E_1) → a_1
  53. Segment(D_1, R) → c_1
  54. Segment(R, H) → b_1
  55. OrthogonalLine(S, g) → d_1
  56. OrthogonalLine(T, h) → e_1
  57. OrthogonalLine(U, f) → f_2
  58. Segment(B, T) → g_2
  59. Segment(A, S) → h_2
  60. Segment(U, D) → i_2
  61. OrthogonalLine(W, q) → j_2
  62. OrthogonalLine(Z, p) → k_2
  63. OrthogonalLine(V, m) → l_2
  64. Segment(W, Z) → m_2
  65. Segment(V, Z) → n_2
  66. Intersect(k_2, m) → F_1
  67. Segment(Z, F_1) → p_2
  68. OrthogonalLine(B_1, r) → q_2
  69. OrthogonalLine(C_1, s) → r_2
  70. OrthogonalLine(A_1, t) → s_2
  71. Intersect(k_2, q) → G_1
  72. Segment(Z, G_1) → t_2
  73. Intersect(r_2, t) → H_1
  74. Intersect(q_2, t) → I_1
  75. Intersect(r, s_2) → J_1
  76. Intersect(r_2, r) → K_1
  77. Intersect(q_2, s) → L_1
  78. Intersect(s_2, s) → M_1
  79. Segment(C_1, H_1) → a_2
  80. Segment(I_1, B_1) → b_2
  81. Segment(J_1, A_1) → d_2
  82. Segment(K_1, C_1) → e_2
  83. Segment(A_1, M_1) → g_3
  84. Segment(K_1, C_1) → c_2
  85. Segment(B_1, L_1) → h_3
  86. Intersect(q_2, r_2) → N_1
  87. Segment(B_1, N_1) → f_3
  88. Segment(J_1, N_1) → i_3
  89. Segment(C_1, N_1) → j_3
  90. Segment(M_1, N_1) → k_3
  91. Intersect(r_1, s_1) → O_1
  92. Segment(P, O_1) → l_3
  93. Segment(H, O_1) → m_3
  94. Segment(Q, O_1) → n_3
  95. Intersect(g_2, h_2) → P_1
  96. AngularBisector(j, l) → p_3, q_3
  97. AngularBisector(j, i) → r_3, s_3
  98. AngularBisector(l, i) → t_3, a_3
  99. Intersect(a_3, j) → Q_1
  100. Intersect(r_3, l) → R_1
  101. Intersect(p_3, i) → S_1
  102. Segment(F, Q_1) → b_3
  103. Segment(E, R_1) → c_3
  104. Segment(H, S_1) → d_3
  105. Intersect(c_3, d_3) → T_1
  106. AngularBisector(B, D, A) → e_3
  107. AngularBisector(D, A, B) → f_4
  108. AngularBisector(A, B, D) → g_4
  109. Intersect(f_4, g) → U_1
  110. Intersect(g_4, h) → V_1
  111. Intersect(e_3, f) → W_1
  112. Segment(A, U_1) → h_4
  113. Segment(B, V_1) → i_4
  114. Segment(D, W_1) → j_4
  115. Intersect(j_4, h_4) → Z_1
  116. AngularBisector(J, L, K) → k_4
  117. AngularBisector(L, J, K) → l_4
  118. AngularBisector(L, K, J) → m_4
  119. Intersect(m_4, q) → A_2
  120. Intersect(l_4, p) → B_2
  121. Intersect(k_4, m) → C_2
  122. Segment(L, C_2) → n_4
  123. Segment(J, B_2) → p_4
  124. Segment(K, A_2) → q_4
  125. AngularBisector(O, M, N) → r_4
  126. AngularBisector(M, N, O) → s_4
  127. AngularBisector(N, O, M) → t_4
  128. Intersect(s_4, t) → D_2
  129. Intersect(r_4, s) → E_2
  130. Intersect(t_4, r) → F_2
  131. Segment(O, F_2) → a_4
  132. Segment(N, D_2) → b_4
  133. Segment(M, E_2) → c_4
  134. Intersect(a_4, c_4) → G_2
  135. Intersect(p_4, n_4) → H_2
  136. OrthogonalLine(H, i) → d_4
  137. OrthogonalLine(E, l) → e_4
  138. OrthogonalLine(F, j) → f_5
  139. Intersect(f_5, e_4) → I_2
  140. Segment(F, I_2) → g_5
  141. Segment(E, I_2) → h_5
  142. Segment(R, I_2) → i_5
  143. OrthogonalLine(B, h) → j_5
  144. OrthogonalLine(A, g) → k_5
  145. OrthogonalLine(D, f) → l_5
  146. Intersect(k_5, g) → J_2
  147. Intersect(l_5, f) → K_2
  148. Intersect(j_5, h) → L_2
  149. Segment(D, K_2) → m_5
  150. Segment(B, L_2) → n_5
  151. Segment(A, J_2) → p_5
  152. Segment(L, J) → q_5
  153. Segment(J, K) → r_5
  154. OrthogonalLine(J, p) → s_5
  155. Intersect(s_5, p) → M_2
  156. Segment(J, M_2) → t_5
  157. OrthogonalLine(N, t) → a_5
  158. OrthogonalLine(O, r) → b_5
  159. OrthogonalLine(M, s) → c_5
  160. Intersect(b_5, a_5) → N_2
  161. Intersect(b_5, r) → O_2
  162. Intersect(c_5, s) → P_2
  163. Intersect(a_5, t) → Q_2
  164. Segment(O, N_2) → d_5
  165. Segment(O, O_2) → e_5
  166. Segment(M, N_2) → f_6
  167. Segment(M, P_2) → g_6
  168. Segment(Q_2, N_2) → h_6
  169. Segment(Q_2, N) → i_6
  170. Intersect(m_5, p_5) → R_2
  171. Intersect(a, f_1) → S_2
  172. Intersect(i_1, g_1) → T_2
  173. Intersect(k_1, l_1) → U_2
  174. Intersect(n_1, p_1) → V_2
  175. Intersect(m_2, n_2) → W_2

AI 绘图提示语参考

绘制一个包含多种特殊三角形的几何探究图。先构造一个等边三角形，再构造一个任意三角形，分别作出它们的重心、垂心、内心和外心。对于每个三角形，连接顶点与对边中点得到中线，三条中线交于重心；过顶点作对边的垂线，三条垂线交于垂心；作各角的角平分线，交于内心；作各边的垂直平分线，交于外心。最后，用不同颜色或线型区分这些特殊点及其连线，以便观察它们是否共点。
以上提示语为根据原始绘图命令整理而来，仅供参考。