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七边形内角与外角和定理演示

七边形内角与外角和定理演示
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2015-10-29 01:00
多边形角几何作图滑块动画
所属专题 Polygons & Angles Polygons & Angles Polygons & Angles ( s n attar) Polygons & Angles Polygons & Angles Polygons & Angles Polygons & Angles Geometry: Polygons & Angles Geometry: Polygons & Angles
①本范例通过交互式演示，向学生展示任意七边形的内角和为180×5=900度，以及外角和为360度的几何现象；②涉及的核心数学概念包括多边形内角和定理、外角和定理、角度计算；③作图要点——从下方『构造摘要』提炼用到了Ray（射线）、Circle（圆）、Intersect（相交）、Segment（线段）、CircleSector（扇形）、Vector（向量）、Translate（平移）、Mirror（镜像）、Dilate（缩放）、If（条件）、Rotate（旋转）、Polygon（多边形）、Point（点）等GeoGebra命令/对象；④适合高中阶段学生学习和理解多边形几何性质，也可用于数学教学演示。

GeoGebra 命令及参数

  1. Ray(N_1, D) → l_4
  2. Circle(D, 1.5) → k
  3. Intersect(k, l_4) → W_1, Z_1
  4. Ray(D, A_1) → a_3
  5. Intersect(k, a_3, 1) → D_1
  6. Segment(C, A_1) → i_3
  7. Circle(C, 1.5) → h
  8. Intersect(h, i_3) → L,
  9. Ray(C, D) → c
  10. Intersect(h, c, 1) → P
  11. Ray(A, B) → a
  12. Ray(B, C) → b
  13. Ray(E, A) → e
  14. Circle(A, 1.5) → f
  15. Circle(B, 1.5) → g
  16. Circle(E, 1.5) → p
  17. Segment(C, A) → i
  18. Segment(C, E) → j
  19. Intersect(f, a, 1) → F
  20. Intersect(f, i, 1) → G
  21. Intersect(f, e, 1) → H
  22. Intersect(f, e, 2) → I
  23. Intersect(p, e) → J,
  24. Intersect(p, j) → K,
  25. Intersect(k, c, 2) → O
  26. Intersect(h, j, 1) → Q
  27. Intersect(h, i, 1) → R
  28. Intersect(h, b, 1) → S
  29. Intersect(g, a, 2) → T
  30. Intersect(g, b, 1) → U
  31. Intersect(g, a, 1) → V
  32. CircleSector(A, F, G) → q
  33. CircleSector(A, G, H) → r
  34. CircleSector(B, U, V) → s
  35. CircleSector(C, Q, R) → c_1
  36. CircleSector(C, R, S) → d_1
  37. Intersect(k, c, 1) → W
  38. CircleSector(E, J, K) → h_1
  39. CircleSector(A, I, F) → p_1
  40. CircleSector(B, T, U) → q_1
  41. Intersect(h, b, 2) → Z
  42. Vector(A, B) → u
  43. Vector(C, B) → v
  44. Vector(E, D) → w
  45. Vector(C, D) → u_1
  46. Vector(C, E) → v_1
  47. Vector(A, E) → w_1
  48. Translate(q, u) → q'
  49. Translate(d_1, v) → s_1
  50. Mirror(s_1, B) → t_1
  51. Dilate(A, If[0 < l ≤ 1, 1 - l, 0], B) → A'
  52. Dilate(C, If[0 < l ≤ 1, 1 - l, 0], B) → C'
  53. Dilate(A, If[1 < l ≤ 2, 2 - l, 0], E) → A'_1
  54. Dilate(C, If[1 < l ≤ 2, 2 - l, 0], E) → C'_1
  55. Dilate(C, If[2 < l ≤ 3, 3 - l, 0], D) → C'_2
  56. Dilate(E, If[2 < l ≤ 3, 3 - l, 0], D) → E'
  57. Vector(A, B) → a_1
  58. Vector(C, C') → b_1
  59. Vector(A, A'_1) → i_1
  60. Vector(E, E') → j_1
  61. Vector(C, C'_2) → l_1
  62. Vector(C, C'_1) → m_1
  63. Translate(d_1, b_1) → c_2
  64. Translate(r, i_1) → r'_1
  65. Translate(r, w_1) → r'_2
  66. Translate(c_1, v_1) → f_2
  67. Translate(c_1, m_1) → g_2
  68. CircleSector(C, P, L) → t
  69. Translate(t, u_1) → t'
  70. Translate(t, l_1) → t'_1
  71. Translate(q, a_1) → r'
  72. Translate(d_1, b_1) → h_2
  73. If(0 < l < 1, Rotate[h_2, ((l * 180))°, C']) → k_2
  74. Vector(A, A') → n_1
  75. Translate(q, n_1) → q'_1
  76. Mirror(r'_2, E) → p_2
  77. If(1 < l < 2, Rotate[r'_1, (((l - 1) * 180))°, A'_1]) → q_2
  78. Ray(C, E) → i_2
  79. Vector(D, C) → u_2
  80. Vector(E, C) → v_2
  81. Vector(A, C) → w_2
  82. Vector(B, C) → l_2
  83. Dilate(B, If[0 < o < 1, 1 - o, 0], C) → B'
  84. Dilate(A, If[1 < o < 2, 2 - o, 0], C) → A'_2
  85. Dilate(E, If[2 < o < 3, 3 - o, 0], C) → E'_1
  86. Dilate(D, If[3 < o < 4, 4 - o, 0], C) → D'
  87. Vector(B, B') → m_2
  88. Vector(A, A'_2) → n_2
  89. Vector(E, E'_1) → s_2
  90. Vector(D, D') → t_2
  91. Translate(q_1, m_2) → c_3
  92. Translate(q_1, l_2) → d_3
  93. Translate(p_1, w_2) → e_3
  94. If(1 < o < 2, Translate[p_1, n_2]) → h_3
  95. Ray(A_1, E) → b_3
  96. Circle(A_1, 1.5) → q_3
  97. Intersect(q_3, i_3) → M,
  98. Intersect(q_3, b_3, 1) → N
  99. Intersect(q_3, a_3, 2) → B_1
  100. Intersect(k, c, 2) → C_1
  101. Intersect(k, c, 1) → E_1
  102. CircleSector(C, L, Q) → d
  103. Intersect(p, b_3, 1) → F_1
  104. Intersect(p, i_2) → G_1, H_2
  105. CircleSector(E, G_1, N) → e_1
  106. Dilate(E, If[2 < l < 3, 3 - l, 0], A_1) → E'_2
  107. Dilate(C, If[2 < l < 3, 3 - l, 0], A_1) → C'_3
  108. Dilate(C, If[3 < l < 4, 4 - l, 0], D) → C'_4
  109. Dilate(A_1, If[3 < l < 4, 4 - l, 0], D) → H_1
  110. Vector(E, A_1) → f_1
  111. Vector(C, A_1) → g_1
  112. Translate(e_1, f_1) → k_1
  113. Translate(d, g_1) → d'
  114. Vector(E, E'_2) → a_2
  115. Vector(C, C'_3) → b_2
  116. Translate(e_1, a_2) → d_2
  117. Translate(d, b_2) → d'_1
  118. Rotate(d_2, If[2 < l < 3, (((l - 2) * 180))°, 0], E'_2) → e_2
  119. Mirror(k_1, A_1) → r_2
  120. CircleSector(A_1, N, M) → f_3
  121. Intersect(q_3, a_3, 1) → I_1
  122. CircleSector(A_1, M, I_1) → g_3
  123. CircleSector(D, D_1, E_1) → k_3
  124. Vector(C, D) → j_2
  125. Vector(A_1, D) → u_3
  126. Translate(g_3, u_3) → p_3
  127. Translate(t, j_2) → t'_2
  128. Mirror(p_3, D) → r_3
  129. Vector(A_1, H_1) → v_3
  130. Vector(C, C'_4) → w_3
  131. Translate(g_3, v_3) → s_3
  132. Translate(t, w_3) → t'_3
  133. Rotate(s_3, If[3 < l < 4, (((l - 3) * 180))°, 0], H_1) → t_3
  134. Intersect(p, b_3, 2) → J_1
  135. Polygon(A, B, C) → poly1, c_4, a_4, b_4
  136. Polygon(A, C, E) → poly2, e_4, a_5, c_5
  137. Polygon(E, C, A_1) → poly3, a_6, e_5, c_6
  138. Polygon(A_1, C, D) → poly4, d_4, a_7, c_7
  139. Intersect(p, e, 1) → L_1
  140. CircleSector(E, J_1, L_1) → f_4
  141. CircleSector(A_1, B_1, N) → g_4
  142. Vector(E, C) → j_3
  143. Vector(A_1, C) → l_3
  144. Vector(D, C) → m_3
  145. Translate(f_4, j_3) → k_4
  146. Translate(g_4, l_3) → p_4
  147. CircleSector(D, Z_1, D_1) → h_4
  148. Translate(h_4, m_3) → q_4
  149. Dilate(E, If[2 < o < 3, 3 - o, 0], C) → E'_3
  150. Dilate(A_1, If[3 < o < 4, 4 - o, 0], C) → M_1
  151. Dilate(D, If[4 < o < 5, 5 - o, 0], C) → D'_1
  152. Vector(E, E'_3) → n_3
  153. Vector(A_1, M_1) → u_4
  154. Vector(D, D'_1) → v_4
  155. If(2 < o < 3, Translate[f_4, n_3]) → r_4
  156. If(3 < o < 4, Translate[g_4, u_4]) → s_4
  157. If(4 < o < 5, Translate[h_4, v_4]) → t_4
  158. Ray(C, N_1) → i_4
  159. Circle(N_1, 1.5) → d_5
  160. Intersect(d_5, i_4, 2) → K_1
  161. Intersect(d_5, l_4, 1) → O_1
  162. Intersect(d_5, i_4, 1) → P_1
  163. Intersect(h, i_4, 1) → Q_1
  164. CircleSector(C, Q_1, P) → f_5
  165. CircleSector(N_1, O_1, P_1) → g_5
  166. Intersect(k, l_4, 1) → R_1
  167. CircleSector(D, E_1, R_1) → h_5
  168. Dilate(D, If[4 < l < 5, 5 - l, 0], N_1) → D'_2
  169. Dilate(C, If[4 < l < 5, 5 - l, 0], N_1) → C'_5
  170. Vector(C, N_1) → w_4
  171. Vector(D, N_1) → j_4
  172. Translate(f_5, w_4) → k_5
  173. Translate(h_5, j_4) → p_5
  174. Mirror(p_5, N_1) → q_5
  175. Vector(D, D'_2) → m_4
  176. Vector(C, C'_5) → n_4
  177. Translate(h_5, m_4) → r_5
  178. Translate(f_5, n_4) → s_5
  179. Rotate(r_5, If[4 < l < 5, (((l - 4) * 180))°, 0], D'_2) → t_5
  180. Polygon(D, C, N_1) → poly5, n_5, d_6, c_8
  181. Segment(D, C) → b_5
  182. Ray(C, E) → i_5
  183. Ray(C, A_1) → j_5
  184. Ray(C, D) → l_5
  185. Point(j_5) → T_1
  186. Point(l_5) → U_1
  187. Ray(A_1, T_1) → m_5
  188. Ray(D, U_1) → b_6
  189. Polygon(A_1, E, A, B, C, N_1, D) → poly6, a_8, e_6, a_9, b_7, c_9, n_6, d_7
  190. Intersect(d_5, l_4, 1) → S_1
  191. Intersect(d_5, i_4, 2) → V_1
  192. CircleSector(N_1, V_1, S_1) → f_6
  193. CircleSector(C, Z, Q_1) → r_1
  194. Vector(N_1, C) → u_5
  195. Translate(f_6, u_5) → g_6
  196. Dilate(N_1, If[5 < o < 6, 6 - o, 0], C) → A_2
  197. Vector(N_1, A_2) → v_5
  198. Translate(f_6, v_5) → h_6
  199. Vector(D, C) → w_5
  200. Translate(h_4, w_5) → k_6