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多边形外角和的无需言语证明

多边形外角和的无需言语证明
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2020-01-19 18:03
多边形角三角函数几何作图
①本范例通过动态演示证明多边形外角和恒为360度；②涉及核心数学概念：多边形、外角、旋转、向量；③作图要点：使用GeoGebra命令Circle（圆）、Point（点）、Polygon（多边形）、Vector（向量）、Translate（平移）、Rotate（旋转）等构建图形并演示旋转关系；④适合高中数学教学，用于直观理解多边形外角和定理。

GeoGebra 命令及参数

  1. Circle(AA, Taille - 0.6) → c
  2. Point(c) → B_1
  3. Point(c) → C_1
  4. Point(c) → A_1
  5. Polygon(A_1, B_1, C_1) → poly1, c_1, a, b
  6. Centroid(poly1) → D
  7. Circle(AA, Taille + 2) → g
  8. Vector(A_1, D) → u
  9. Vector(C_1, D) → v
  10. Vector(B_1, D) → w
  11. Translate(A_1, Vector[(Voir * u)]) → A
  12. Translate(B_1, Vector[(Voir * w)]) → B
  13. Translate(C_1, Vector[(Voir * v)]) → C
  14. Polygon(A, B, C) → poly2, c_2, a_1, b_1
  15. Line(A, B) → d
  16. Line(B, C) → e
  17. Line(A, C) → f
  18. Intersect(g, e) → E_3, F_3
  19. Intersect(g, f) → G_3, H_4
  20. Intersect(g, d) → I_3, J_3
  21. Segment(E_3, F_3) → h
  22. Segment(I_3, J_3) → i
  23. Angle(C, A, I_3) → α
  24. Angle(B, C, H_4) → β
  25. Angle(A, B, E_3) → γ
  26. Line(D, i) → j
  27. Intersect(c, j) → K_3, L_3
  28. Rotate(K_3, (-α), D) → K'
  29. Segment(D, K') → l
  30. Segment(K_3, D) → m
  31. Rotate(K_3, γ, D) → K'_1
  32. Segment(K'_1, D) → n_5
  33. Segment(G_3, H_4) → p
  34. Angle(K', D, K_3) → δ
  35. Angle(K'_1, D, K') → ε
  36. Angle(K_3, D, K'_1) → ζ
  37. Circle(M_3, Taille) → q
  38. Point(q) → N_2
  39. Point(q) → O_1
  40. Point(q) → P_1
  41. Point(q) → Q_1
  42. Polygon(O_1, P_1, Q_1, N_2) → poly3, o, p_1, q_1, n_1
  43. Centroid(poly3) → S_2
  44. Vector(Q_1, S_2) → z
  45. Vector(P_1, S_2) → r
  46. Vector(N_2, S_2) → s
  47. Vector(O_1, S_2) → t
  48. Translate(N_2, Vector[(Voir * s)]) → N
  49. Translate(O_1, Vector[(Voir * t)]) → O
  50. Translate(P_1, Vector[(Voir * r)]) → P
  51. Translate(Q_1, Vector[(Voir * z)]) → Q
  52. Polygon(N, O, P, Q) → poly4, n', o', p', q'
  53. Line(Q, P) → d_1
  54. Line(P, O) → e_1
  55. Line(O, N) → f_1
  56. Line(Q, N) → g_1
  57. Circle(M_3, Taille + 2) → h_1
  58. Intersect(h_1, d_1) → T_2, U_2
  59. Intersect(h_1, g_1) → V_2, W_2
  60. Intersect(h_1, e_1) → Z_3, D_1
  61. Intersect(h_1, f_1) → E_1, F_1
  62. Segment(T_2, U_2) → i_1
  63. Segment(Z_3, D_1) → j_1
  64. Segment(F_1, E_1) → k_1
  65. Segment(V_2, W_2) → l_1
  66. Angle(U_2, Q, V_2) → η
  67. Angle(D_1, P, U_2) → θ
  68. Angle(F_1, O, D_1) → ι
  69. Angle(V_2, N, F_1) → κ
  70. Line(S_2, l_1) → m_1
  71. Line(S_2, i_1) → r_1
  72. Intersect(h_1, m_1) → G_1, H_1
  73. Segment(G_1, S_2) → s_1
  74. Intersect(h_1, r_1) → I_1, J_1
  75. Segment(S_2, J_1) → t_1
  76. Line(S_2, j_1) → a_2
  77. Intersect(h_1, a_2) → K_1, L_1
  78. Segment(S_2, L_1) → b_2
  79. Line(S_2, k_1) → d_2
  80. Intersect(h_1, d_2) → M_1, N_1
  81. Segment(S_2, M_1) → e_2
  82. Circle(S_2, Taille + 0.4) → g_2
  83. Circle(R_3, Taille) → h_2
  84. Point(h_2) → S_1
  85. Point(h_2) → T_1
  86. Point(h_2) → U_1
  87. Point(h_2) → V_1
  88. Point(h_2) → W_1
  89. Polygon(T_1, U_1, V_1, W_1, S_1) → poly5, t_2, u_1, v_1, w_1, s_2
  90. Centroid(poly5) → R
  91. Vector(S_1, R) → z_1
  92. Vector(T_1, R) → u_2
  93. Vector(U_1, R) → v_2
  94. Vector(V_1, R) → w_2
  95. Vector(W_1, R) → z_2
  96. Translate(S_1, Vector[(Voir * z_1)]) → S
  97. Translate(T_1, Vector[(Voir * u_2)]) → T
  98. Translate(U_1, Vector[(Voir * v_2)]) → U
  99. Translate(V_1, Vector[(Voir * w_2)]) → V
  100. Translate(W_1, Vector[(Voir * z_2)]) → W
  101. Line(V, U) → i_2
  102. Line(U, T) → j_2
  103. Line(T, S) → k_2
  104. Line(S, W) → l_2
  105. Line(V, W) → m_2
  106. Circle(R, Taille + 2) → p_2
  107. Intersect(p_2, m_2) → R_1, Z_1
  108. Intersect(p_2, k_2) → A_2, B_2
  109. Intersect(p_2, i_2) → C_2, D_2
  110. Intersect(p_2, j_2) → E_2, F_2
  111. Intersect(p_2, l_2) → G_2, H_2
  112. Segment(C_2, D_2) → n_2
  113. Segment(E_2, F_2) → q_2
  114. Segment(Z_1, R_1) → r_2
  115. Segment(B_2, A_2) → a_3
  116. Segment(H_2, G_2) → b_3
  117. Angle(V, W, H_2) → λ
  118. Angle(D_2, V, R_1) → μ
  119. Angle(F_2, U, D_2) → ν
  120. Angle(B_2, T, F_2) → ξ
  121. Angle(H_2, S, B_2) → ο
  122. Line(R, r_2) → e_3
  123. Intersect(p_2, e_3) → K_2, L_2
  124. Segment(R, L_2) → f_3
  125. Line(R, n_2) → g_3
  126. Intersect(p_2, g_3) → M_2, O_2
  127. Segment(R, O_2) → h_3
  128. Line(R, a_3) → k_3
  129. Intersect(p_2, k_3) → R_2, Z_2
  130. Segment(R_2, R) → l_3
  131. Polygon(S, T, U, V, W) → poly6, s_3, t_3, u_3, v_3, w_3
  132. Line(R, b_3) → c_3
  133. Intersect(p_2, c_3) → I_2, J_2
  134. Segment(I_2, R) → d_3
  135. Angle(L_2, R, I_2) → ρ
  136. Line(R, q_2) → i_3
  137. Intersect(p_2, i_3) → P_2, Q_2
  138. Segment(R, Q_2) → j_3
  139. Circle(D, Taille - 0.4) → f_2
  140. Circle(A_3, Taille) → p_3
  141. Point(p_3) → H
  142. Point(p_3) → I
  143. Point(p_3) → J
  144. Point(p_3) → K
  145. Point(p_3) → L
  146. Point(p_3) → M
  147. Point(p_3) → E
  148. Point(p_3) → F
  149. Point(p_3) → G
  150. Polygon(E, F, G, H, I, J, K, L, M) → poly7, e_4, f_4, g_4, h_4, i_4, j_4, k_4, l_4, m_3
  151. Centroid(poly7) → Z
  152. Sequence(Vector[Element[liste1, i], Z], i, 1, Length[liste1]) → liste2
  153. Sequence(Translate[Element[liste1, i], Vector[(Voir * Element[liste2, i])]], i, 1, Length[liste1]) → liste3
  154. Polygon(liste3) → poly8
  155. Circle(Z, Taille + 2) → p_4
  156. Sequence(Line[Element[liste3, i], Element[liste3, i + 1]], i, 1, Length[liste3]) → liste4
  157. Sequence(Intersect[p_4, Element[liste4, i], 2], i, 1, Length[liste4]) → liste5
  158. Sequence(Segment[Element[liste5, i], Element[liste3, i]], i, 1, Length[liste3]) → liste6
  159. Line(M, E) → q_4
  160. Line(Element[liste3, 1], Element[liste3, 9]) → r_4
  161. Intersect(p_4, r_4) → B_3, C_3
  162. Segment(Element[liste3, 9], B_3) → s_4
  163. Sequence(Angle[Element[liste5, i], Element[liste3, i + 1], Element[liste5, i + 1]], i, 1, Length[liste5]) → liste7
  164. Angle(B_3, Element[liste3, 1], Element[liste5, 1]) → σ
  165. Angle(Element[liste5, 8], Element[liste3, 9], B_3) → τ
  166. Segment(Element[liste3, 1], B_3) → t_4
  167. Sequence(Rotate[H_3, i, D_3], i, 0, 6.28319, (2 * π / n)) → liste8
  168. Polygon(liste8) → poly9
  169. Sequence(Line[Element[liste8, i], Element[liste8, i + 1]], i, 1, Length[liste8]) → liste9
  170. Sequence(Vector[Element[liste8, i], D_3], i, 1, Length[liste8]) → liste10
  171. Sequence(Translate[Element[liste8, i], Vector[(Voir * Element[liste10, i])]], i, 1, Length[liste8]) → liste11
  172. Circle(D_3, Taille + 2) → k
  173. Sequence(Line[Element[liste11, i], Element[liste11, i + 1]], i, 1, Length[liste11]) → liste12
  174. Sequence(Intersect[k, Element[liste12, i], 2], i, 1, Length[liste12]) → liste13
  175. Polygon(liste11) → poly10
  176. Sequence(Segment[Element[liste13, i], Element[liste11, i]], i, 1, Length[liste11]) → liste14
  177. Sequence(Angle[Element[liste13, i], Element[liste11, i + 1], Element[liste13, i + 1]], i, 1, Length[liste13]) → liste15
  178. Polygon(liste11) → poly11
  179. Element(liste13, 1) → RR
  180. Element(liste13, Length[liste13] - 1) → SS
  181. Element(liste11, 1) → TT
  182. Angle(SS, TT, RR) → υ