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动画演示三角形重心、垂心与特殊点

动画演示三角形重心、垂心与特殊点
几何画板及绘图内容加载中，请稍等……
2016-06-26 09:32
测试改名0向量三角函数解析几何几何作图
①本范例通过动画演示三角形的重心、垂心等特殊点的动态变化与相互关系；②涉及核心数学概念包括：三角形重心、垂心、特殊点、解析几何、几何作图；③作图要点：运用了Line（线）、Segment（线段）、Midpoint（中点）、OrthogonalLine（垂线）、Intersect（交点）、Min（最小值）、Circle（圆）、Dilate（缩放）、Rotate（旋转）、Polygon（多边形）、Point（点）、Ray（射线）、CircleSector（扇形）、Angle（角度）、CircleArc（圆弧）、Vector（向量）、Translate（平移）等GeoGebra命令与对象；④适合高中数学教学，用于理解三角形特殊点的性质与相互联系。

GeoGebra 命令及参数

  1. Line(C, B) → l_1
  2. Segment(B, C) → g
  3. Midpoint(g) → F_3
  4. OrthogonalLine(F_3, l_1) → j_3
  5. Line(A, C) → f_1
  6. Segment(C, A) → h
  7. Midpoint(h) → E_3
  8. OrthogonalLine(E_3, f_1) → k_3
  9. Intersect(k_3, j_3) → I_2
  10. Midpoint(I_2, C) → I_3
  11. Segment(A, B) → f
  12. Midpoint(f) → D_3
  13. Min(list1) → a
  14. Circle(D_3, (0.15 * a)) → c
  15. Circle(E_3, (0.15 * a)) → d
  16. Circle(F_3, (0.15 * a)) → e
  17. Intersect(c, f, 2) → H
  18. Intersect(d, h, 2) → K
  19. Intersect(e, g, 2) → M
  20. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], D_3) → A'
  21. Dilate(B, If[2 < b ≤ 3, 3 - b, 0], F_3) → B'
  22. Dilate(C, If[4 < b ≤ 5, 5 - b, 0], E_3) → C'
  23. Segment(A, A') → l
  24. Segment(B, B') → m
  25. Segment(C, C') → n
  26. Rotate(A, If[1 < b ≤ 2, ((b - 1) * π), π], D_3) → A'_1
  27. Rotate(A', If[1 < b ≤ 2, ((b - 1) * π), π], D_3) → A''
  28. Segment(A'_1, A'') → l'
  29. Rotate(B, If[3 < b ≤ 4, ((b - 3) * π), π], F_3) → B'_1
  30. Rotate(B', If[3 < b ≤ 4, ((b - 3) * π), π], F_3) → B''
  31. Segment(B'_1, B'') → m'
  32. Rotate(C, If[5 < b ≤ 6, ((b - 5) * π), π], E_3) → C'_1
  33. Rotate(C', If[5 < b ≤ 6, ((b - 5) * π), π], E_3) → C''
  34. Segment(C'_1, C'') → n'
  35. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  36. Line(A, B) → i
  37. Point(i) → C_3
  38. Point(i) → G
  39. OrthogonalLine(C, i) → j
  40. Intersect(i, j) → I
  41. Ray(A, G) → k
  42. Ray(B, C_3) → p
  43. Point(k) → J
  44. Point(p) → L
  45. Circle(I, (0.15 * a)) → s
  46. Intersect(s, i, 2) → N
  47. Intersect(s, j, 2) → O_3
  48. CircleSector(I, N, O_3) → t
  49. Angle(C, B, A) → α
  50. Angle(B, A, C) → β
  51. Angle(A, C, B) → γ
  52. Point(f_1) → P
  53. Point(f_1) → Q
  54. Ray(A, P) → g_1
  55. Ray(C, Q) → h_1
  56. Point(g_1) → R
  57. Point(h_1) → S
  58. OrthogonalLine(B, f_1) → k_1
  59. Intersect(f_1, k_1) → T
  60. Circle(T, (0.15 * a)) → d_1
  61. Intersect(d_1, f_1, 1) → U
  62. Intersect(d_1, k_1, 1) → V
  63. CircleSector(T, U, V) → e_1
  64. OrthogonalLine(A, l_1) → m_1
  65. Intersect(l_1, m_1) → W
  66. Circle(W, (0.15 * 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, B_1) → n_1
  73. Ray(B, D_1) → r_1
  74. Point(n_1) → E_1
  75. Point(r_1) → F_1
  76. Dilate(C, If[6 < b ≤ 7, 7 - b, 0], I) → G_1
  77. Segment(C, G_1) → f_2
  78. Dilate(B, If[7 < b ≤ 8, 8 - b, 0], T) → B'_2
  79. Segment(B, B'_2) → g_2
  80. Dilate(A, If[8 < b ≤ 9, 9 - b, 0], W) → A'_2
  81. Segment(A, A'_2) → h_2
  82. Intersect(k_1, m_1) → O
  83. Dilate(I, If[12 < b ≤ 13, 13 - b, 0], O) → I'
  84. Dilate(A, If[12 < b ≤ 13, 13 - b, 0], O) → A'_3
  85. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], O) → T'
  86. Segment(I, I') → i_2
  87. Segment(A, A'_3) → j_2
  88. Segment(T, T') → k_2
  89. Dilate(C, If[12 < b ≤ 13, 13 - b, 0], O) → I_1
  90. Dilate(W, If[12 < b ≤ 13, 13 - b, 0], O) → W'
  91. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], O) → T'_1
  92. Segment(W, W') → l_2
  93. Segment(C, I_1) → m_2
  94. Segment(T, T'_1) → n_2
  95. Dilate(I, If[12 < b ≤ 13, 13 - b, 0], O) → I'_1
  96. Dilate(B, If[12 < b ≤ 13, 13 - b, 0], O) → B'_3
  97. Dilate(W, If[12 < b ≤ 13, 13 - b, 0], O) → W'_1
  98. Segment(B, B'_3) → p_2
  99. Segment(W, W'_1) → q_2
  100. Segment(I, I'_1) → r_2
  101. Midpoint(O, C) → F
  102. Midpoint(O, A) → D
  103. Midpoint(O, B) → E
  104. Dilate(O, If[14 < b ≤ 15, 15 - b, 0], F) → M_1
  105. Dilate(F, If[15 < b ≤ 16, 16 - b, 0], C) → N_1
  106. Segment(F, N_1) → t_2
  107. Rotate(O, If[16 < b ≤ 17, ((b - 16) * π), π], F) → O_1
  108. Rotate(M_1, If[16 < b ≤ 17, ((b - 16) * π), π], F) → P_1
  109. Dilate(O, If[17 < b ≤ 18, 18 - b, 0], E) → Q_1
  110. Dilate(E, If[18 < b ≤ 19, 19 - b, 0], B) → R_1
  111. Segment(E, R_1) → e_2
  112. Rotate(O, If[19 < b ≤ 20, ((b - 19) * π), π], E) → S_1
  113. Rotate(Q_1, If[19 < b ≤ 20, ((b - 19) * π), π], E) → T_1
  114. Dilate(O, If[20 < b ≤ 21, 21 - b, 0], D) → U_1
  115. Dilate(D, If[21 < b ≤ 22, 22 - b, 0], A) → V_1
  116. Segment(D, V_1) → h_3
  117. Rotate(O, If[22 < b ≤ 23, ((b - 22) * π), π], D) → W_1
  118. Rotate(U_1, If[22 < b ≤ 23, ((b - 22) * π), π], D) → Z_1
  119. Intersect(e, l_1) → A_2, B_2
  120. Intersect(e, j_3) → C_2, D_2
  121. CircleSector(F_3, A_2, C_2) → c_3
  122. Circle(E_3, (0.15 * a)) → d_3
  123. Intersect(d_3, k_3) → E_2, F_2
  124. Intersect(d_3, f_1) → G_2, H_2
  125. CircleSector(E_3, G_2, E_2) → e_3
  126. Midpoint(O, I_2) → J_2
  127. Dilate(O, If[28 < b ≤ 29, 29 - b, 0], J_2) → K_2
  128. Dilate(J_2, If[29 < b ≤ 30, 30 - b, 0], I_2) → L_2
  129. Rotate(O, If[30 < b ≤ 31, ((b - 30) * π), π], J_2) → M_2
  130. Rotate(K_2, If[30 < b ≤ 31, ((b - 30) * π), π], J_2) → N_2
  131. Dilate(J_2, If[31 < b ≤ 32, 32 - b, 0], D_3) → O_2
  132. Rotate(J_2, If[32 < b ≤ 34, ((b - 32) * π), (2 * π)], J_2) → P_2
  133. Rotate(O_2, If[32 < b ≤ 34, ((b - 32) * π), (2 * π)], J_2) → Q_2
  134. Segment(P_2, Q_2) → s_3
  135. CircleArc(J_2, D_3, Q_2) → t_3
  136. Circle(J_2, D_3) → c_4
  137. Point(j) → R_2
  138. Point(j) → S_2
  139. Ray(C, R_2) → j_4
  140. Ray(I, S_2) → k_4
  141. Point(j_4) → T_2
  142. Point(k_4) → U_2
  143. Angle(B, T, C) → δ
  144. Angle(C, I, A) → ε
  145. Angle(A, W, B) → ζ
  146. Angle(I_2, E_3, C) → η
  147. Angle(I_2, D_3, A) → θ
  148. Angle(I_2, F_3, B) → ι
  149. Line(I_2, D_3) → s_2
  150. Line(A, W) → a_2
  151. Line(C, I) → b_2
  152. Line(B, T) → c_2
  153. Dilate(I_2, If[0 < v ≤ 1, 1 - v, 0], J_2) → C_1
  154. Dilate(J_2, If[1 < v ≤ 2, 2 - v, 0], O) → H_1
  155. Segment(J_2, H_1) → f_3
  156. Rotate(I_2, If[2 < v ≤ 3, ((v - 2) * π), π], J_2) → K_1
  157. Rotate(C_1, If[2 < v ≤ 3, ((v - 2) * π), π], J_2) → V_2
  158. Dilate(J_2, If[3 < v ≤ 4, 4 - v, 0], D_3) → W_2
  159. Segment(J_2, W_2) → i_3
  160. Rotate(J_2, If[4 < v ≤ 6, ((v - 4) * π), (2 * π)], J_2) → Z_2
  161. Rotate(W_2, If[4 < v ≤ 6, ((v - 4) * π), (2 * π)], J_2) → A_3
  162. CircleArc(J_2, D_3, A_3) → p_3
  163. Dilate(I_2, If[6 < v ≤ 7, 7 - v, 0], I_3) → L_1
  164. Dilate(I_3, If[7 < v ≤ 8, 8 - v, 0], C) → B_3
  165. Segment(I_3, B_3) → n_3
  166. Dilate(J_2, If[8 < v ≤ 9, 9 - v, 0], I_2) → G_3
  167. Vector(J_2, G_3) → w_1
  168. Translate(Z_2, w_1) → H_3
  169. Translate(A_3, w_1) → J_3
  170. Segment(H_3, J_3) → q_3
  171. Rotate(H_3, If[9 < v ≤ 10, ((v - 9) * π), π], I_2) → K_3
  172. Rotate(J_3, If[9 < v ≤ 10, ((v - 9) * π), π], I_2) → L_3
  173. Segment(K_3, L_3) → r_3
  174. Rotate(K_3, If[10 < v ≤ 11, ((v - 10) * π), π], I_3) → M_3
  175. Rotate(L_3, If[10 < v ≤ 11, ((v - 10) * π), π], I_3) → N_3
  176. Segment(M_3, N_3) → g_4
  177. Midpoint(I_2, A) → R_3
  178. Midpoint(I_2, B) → S_3
  179. Dilate(I_2, If[6 < v ≤ 7, 7 - v, 0], R_3) → U_3
  180. Dilate(I_2, If[6 < v ≤ 7, 7 - v, 0], S_3) → W_3
  181. Dilate(R_3, If[7 < v ≤ 8, 8 - v, 0], A) → J_1
  182. Dilate(S_3, If[7 < v ≤ 8, 8 - v, 0], B) → B_4
  183. Segment(R_3, J_1) → f_4
  184. Segment(S_3, B_4) → h_4
  185. Dilate(J_2, If[11 < v ≤ 12, 12 - v, 0], I_2) → D_4
  186. Vector(J_2, D_4) → u_1
  187. Translate(Z_2, u_1) → G_4
  188. Translate(A_3, u_1) → H_4
  189. Segment(G_4, H_4) → l_4
  190. Angle(D, J_2, D_3) → μ
  191. Rotate(G_4, If[12 < v ≤ 13, ((-(v - 12)) * μ), (-μ)], I_2) → P_3
  192. Rotate(H_4, If[12 < v ≤ 13, ((-(v - 12)) * μ), (-μ)], I_2) → Q_3
  193. Segment(P_3, Q_3) → m_4
  194. Rotate(P_3, If[13 < v ≤ 14, ((-(v - 13)) * π), (-π)], R_3) → T_3
  195. Rotate(Q_3, If[13 < v ≤ 14, ((-(v - 13)) * π), (-π)], R_3) → V_3
  196. Segment(T_3, V_3) → n_4
  197. Dilate(J_2, If[14 < v ≤ 15, 15 - v, 0], I_2) → Z_3
  198. Vector(J_2, Z_3) → v_1
  199. Translate(Z_2, v_1) → A_4
  200. Translate(A_3, v_1) → C_4
  201. Segment(A_4, C_4) → p_4
  202. Angle(D_3, J_2, E) → κ
  203. Rotate(A_4, If[15 < v ≤ 16, ((v - 15) * κ), κ], I_2) → E_4
  204. Rotate(C_4, If[15 < v ≤ 16, ((v - 15) * κ), κ], I_2) → F_4
  205. Segment(E_4, F_4) → r_4
  206. Rotate(E_4, If[16 < v ≤ 17, ((v - 16) * π), π], S_3) → I_4
  207. Rotate(F_4, If[16 < v ≤ 17, ((v - 16) * π), π], S_3) → J_4
  208. Segment(I_4, J_4) → s_4
  209. Circle(I_2, C) → d_4
  210. Rotate(B, If[17 < v ≤ 19, ((v - 17) * π), (2 * π)], I_2) → B'_4
  211. CircleArc(I_2, B, B'_4) → e_4
  212. Line(B, C) → q
  213. Line(A, C) → r
  214. Line(B, A) → i_1
  215. Segment(A, F_3) → j_1
  216. Segment(B, E_3) → s_1
  217. Segment(C, D_3) → t_1
  218. Intersect(t_1, s_1) → K_4
  219. Segment(K_4, J_2) → d_2
  220. Line(O, I_2) → g_3
  221. Circle(J_2, d_2) → q_4
  222. Intersect(q_4, g_3, 1) → L_4
  223. Circle(L_4, d_2) → t_4
  224. Intersect(t_4, g_3, 1) → M_4
  225. Dilate(K_4, If[0 < w ≤ 1, 1 - w, 0], J_2) → N_4
  226. Segment(K_4, N_4) → l_3
  227. Dilate(J_2, If[1 < w ≤ 2, 2 - w, 0], L_4) → O_4
  228. Segment(J_2, O_4) → m_3
  229. Dilate(L_4, If[2 < w ≤ 3, 3 - w, 0], M_4) → P_4
  230. Segment(L_4, P_4) → a_3
  231. Dilate(M_4, If[3 < w ≤ 4, 4 - w, 0], O) → Q_4
  232. Segment(M_4, Q_4) → b_3
  233. Rotate(K_4, If[4 < w ≤ 5, ((w - 4) * π), π], J_2) → R_4
  234. Rotate(N_4, If[4 < w ≤ 5, ((w - 4) * π), π], J_2) → S_4
  235. Segment(R_4, S_4) → i_4
  236. Rotate(R_4, If[5 < w ≤ 6, ((w - 5) * π), π], L_4) → T_4
  237. Rotate(S_4, If[5 < w ≤ 6, ((w - 5) * π), π], L_4) → U_4
  238. Segment(T_4, U_4) → a_4
  239. Rotate(T_4, If[6 < w ≤ 7, ((w - 6) * π), π], M_4) → V_4
  240. Rotate(U_4, If[6 < w ≤ 7, ((w - 6) * π), π], M_4) → W_4
  241. Segment(V_4, W_4) → b_4
  242. Line(O, I_2) → f_5

AI 绘图提示语参考

画一个三角形，标出其重心、垂心以及相关的特殊点，通过动画展示这些点之间的动态关系。首先画出三角形的三条边，然后找到每条边的中点并画出垂直于边的线，这些线的交点即为垂心。接着，连接三角形的顶点与对边的中点，这些线的交点即为重心。再画出以这些中点为圆心、一定半径为半径的圆，这些圆与边的交点为三角形的特殊点。最后，通过旋转和缩放这些点，展示它们之间的动态变化。
以上提示语为根据原始绘图命令整理而来，仅供参考。