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动画109：四边形与等腰三角形的旋转构造

动画109：四边形与等腰三角形的旋转构造
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2016-09-07 03:11
多边形三角函数变换几何作图滑块动画
所属专题 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2
本范例演示了一个由正方形、四边形及等腰三角形构成的动态几何图形，通过旋转、平移和缩放等变换生成一系列相互关联的图形元素。核心数学概念包括多边形（正方形、四边形、等腰三角形）、三角函数（角度参数）、旋转与平移变换、中点与交点、以及整数参数控制下的动态变化。作图要点：从构造摘要可见，使用了Polygon、Angle、Segment、Midpoint、Rotate、Min、Circle、Intersect、Dilate、CircleSector、Vector、Translate、Line、Point等GeoGebra命令，创建了多个多边形（poly1至poly5）、线段、圆、扇形、向量及点对象，并通过角度α、β、γ和整数参数驱动动画。适合中学几何教学或数学爱好者探索动态几何变换，可用于理解旋转对称性、等腰三角形性质及参数化构造。

GeoGebra 命令及参数

  1. Polygon(A, B, C, D) → poly1, a, b, c, d
  2. Polygon(D, C, 4) → poly2, f, g, h, i, E, F
  3. Polygon(C, B, 4) → poly3, j, k, l, m, G, H
  4. Polygon(B, A, 4) → poly4, n, p, q, r, I, J
  5. Polygon(A, D, 4) → poly5, s, t, e, f_1, K, L
  6. Angle(poly2) → α, β, γ, δ
  7. Angle(poly5) → ε, ζ, η, θ
  8. Angle(poly4) → ι, κ, λ, μ
  9. Angle(poly3) → ν, ξ, ο, ρ
  10. Segment(A, D) → g_1
  11. Segment(D, C) → h_1
  12. Segment(C, B) → i_1
  13. Segment(B, A) → j_1
  14. Midpoint(C, F) → M
  15. Midpoint(D, L) → N
  16. Midpoint(A, J) → O
  17. Midpoint(B, H) → P
  18. Segment(O, M) → k_1
  19. Segment(P, N) → l_1
  20. Segment(D, F) → m_1
  21. Segment(F, E) → n_1
  22. Segment(E, C) → p_1
  23. Segment(C, H) → q_1
  24. Segment(H, G) → r_1
  25. Segment(G, B) → s_1
  26. Segment(A, I) → t_1
  27. Segment(A, L) → a_1
  28. Segment(L, K) → b_1
  29. Segment(K, D) → c_1
  30. Segment(J, I) → d_1
  31. Segment(J, B) → e_1
  32. Rotate(D, If[0 < o ≤ 1, (o * π / 2), π / 2], D) → D'
  33. Rotate(C, If[0 < o ≤ 1, (o * π / 2), π / 2], D) → C'
  34. Segment(D, F) → g_2
  35. Min({g_1, j_1, i_1, h_1}) → u
  36. Circle(D, (0.25 * u)) → c_2
  37. Intersect(c_2, h_1, 1) → Q
  38. Intersect(i, g_2) → R
  39. Intersect(i, g_2) → S
  40. Intersect(c_2, g_2) → T,
  41. Intersect(k_1, l_1) → U
  42. Circle(U, (0.2 * u)) → e_2
  43. Intersect(e_2, l_1, 1) → V
  44. Intersect(e_2, k_1, 2) → W
  45. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], B) → A'
  46. Segment(A, A') → i_2
  47. Dilate(B, If[1 < o ≤ 2, 2 - o, 0], C) → B'
  48. Segment(B, B') → j_2
  49. Dilate(C, If[2 < o ≤ 3, 3 - o, 0], D) → C'_1
  50. Segment(C, C'_1) → k_2
  51. Dilate(D, If[3 < o ≤ 4, 4 - o, 0], A) → D'_1
  52. Segment(D, D'_1) → l_2
  53. Polygon(A, B, C, D) → poly6, a_2, b_2, c_3, d_3
  54. Circle(C, (0.2 * u)) → d_2
  55. Segment(C, E) → f_2
  56. Segment(C, D) → m_2
  57. Intersect(d_2, m_2) → Z,
  58. Intersect(d_2, f_2) → A_1,
  59. CircleSector(C, A_1, Z) → p_2
  60. Rotate(C, If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], C) → C'_2
  61. Rotate(C'_1, If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], C) → B_1
  62. Segment(C'_2, B_1) → n_2
  63. Rotate(B, If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], B) → B'_1
  64. Rotate(B', If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], B) → B''
  65. Segment(B'_1, B'') → q_2
  66. Rotate(A, If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], A) → A'_1
  67. Rotate(A', If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], A) → A''
  68. Segment(A'_1, A'') → r_2
  69. Rotate(D, If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], D) → D'_2
  70. Rotate(D'_1, If[4 < o ≤ 5, ((-(o - 4)) * π / 2), ((-π)) / 2], D) → C_1
  71. Segment(D'_2, C_1) → s_2
  72. Rotate(C'_2, If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], E) → D_1
  73. Rotate(B_1, If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], E) → E_1
  74. Segment(D_1, E_1) → t_2
  75. Rotate(B'_1, If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], G) → F_1
  76. Rotate(B'', If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], G) → B'''
  77. Segment(F_1, B''') → f_3
  78. Rotate(A'_1, If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], I) → G_1
  79. Rotate(A'', If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], I) → A'''
  80. Segment(G_1, A''') → g_3
  81. Rotate(D'_2, If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], K) → H_1
  82. Rotate(C_1, If[5 < o ≤ 6, ((-(o - 5)) * π / 2), ((-π)) / 2], K) → I_1
  83. Segment(H_1, I_1) → h_3
  84. Rotate(D_1, If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], F) → J_1
  85. Rotate(E_1, If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], F) → K_1
  86. Segment(J_1, K_1) → i_3
  87. Rotate(F_1, If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], H) → L_1
  88. Rotate(B''', If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], H) → M_1
  89. Segment(L_1, M_1) → j_3
  90. Rotate(G_1, If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], J) → N_1
  91. Rotate(A''', If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], J) → O_1
  92. Segment(N_1, O_1) → k_3
  93. Rotate(H_1, If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], L) → P_1
  94. Rotate(I_1, If[6 < o ≤ 7, ((-(o - 6)) * π / 2), ((-π)) / 2], L) → Q_1
  95. Segment(P_1, Q_1) → l_3
  96. Dilate(C, If[7 < o ≤ 8, 8 - o, 0], E) → C'_3
  97. Dilate(C, If[7 < o ≤ 8, 8 - o, 0], D) → C'_4
  98. Dilate(C, If[7 < o ≤ 8, 8 - o, 0], A) → C'_5
  99. Dilate(C, If[7 < o ≤ 8, 8 - o, 0], B) → C'_6
  100. Vector(C, C'_3) → v
  101. Vector(C, C'_4) → w
  102. Vector(C, C'_5) → u_1
  103. Vector(C, C'_6) → v_1
  104. Translate(p_2, v) → e_3
  105. Translate(p_2, w) → p_3
  106. Translate(p_2, u_1) → q_3
  107. Translate(p_2, v_1) → r_3
  108. Rotate(e_3, If[7 < o ≤ 8, ((o - 7) * π / 2), π / 2], C'_3) → s_3
  109. Angle(poly1) → σ, τ, υ, ϕ
  110. Line(D, C) → m_3
  111. Line(C, B) → n_3
  112. Line(B, A) → t_3
  113. Rotate(p_3, If[7 < o ≤ 8, ((-(o - 7)) * (π + ϕ)), (-(π + ϕ))], C'_4) → c_4
  114. Angle(H, C, E) → χ
  115. Rotate(r_3, If[7 < o ≤ 8, ((-(o - 7)) * χ), (-χ)], C'_6) → d_4
  116. Line(A, n_2) → a_3
  117. Point(a_3) → R_1
  118. Angle(R_1, A, I) → ψ
  119. Rotate(q_3, If[7 < o ≤ 8, ((o - 7) * ψ), ψ], C'_5) → e_4
  120. Dilate(E, If[8 < o ≤ 9, 9 - o, 0], F) → E'
  121. Dilate(B, If[8 < o ≤ 9, 9 - o, 0], G) → B'_2
  122. Dilate(A, If[8 < o ≤ 9, 9 - o, 0], I) → A'_2
  123. Dilate(D, If[8 < o ≤ 9, 9 - o, 0], K) → D'_3
  124. Vector(E, E') → w_1
  125. Vector(D, D'_3) → u_2
  126. Vector(A, A'_2) → v_2
  127. Vector(B, B'_2) → w_2
  128. Translate(s_3, w_1) → f_4
  129. Translate(c_4, u_2) → g_4
  130. Translate(e_4, v_2) → h_4
  131. Translate(d_4, w_2) → k_4
  132. Rotate(f_4, If[8 < o ≤ 9, ((o - 8) * π / 2), π / 2], E') → p_4
  133. Rotate(g_4, If[8 < o ≤ 9, ((o - 8) * π / 2), π / 2], D'_3) → q_4
  134. Rotate(h_4, If[8 < o ≤ 9, ((o - 8) * π / 2), π / 2], A'_2) → r_4
  135. Rotate(k_4, If[8 < o ≤ 9, ((o - 8) * π / 2), π / 2], B'_2) → s_4
  136. Dilate(F, If[9 < o ≤ 10, 10 - o, 0], D) → F'
  137. Dilate(K, If[9 < o ≤ 10, 10 - o, 0], L) → K'
  138. Dilate(I, If[9 < o ≤ 10, 10 - o, 0], J) → I'
  139. Dilate(G, If[9 < o ≤ 10, 10 - o, 0], H) → G'
  140. Vector(K, K') → u_3
  141. Vector(I, I') → v_3
  142. Vector(G, G') → w_3
  143. Vector(F, F') → b_3
  144. Translate(p_4, b_3) → t_4
  145. Translate(q_4, u_3) → c_5
  146. Translate(s_4, w_3) → d_5
  147. Translate(r_4, v_3) → e_5
  148. Rotate(t_4, If[9 < o ≤ 10, ((o - 9) * π / 2), π / 2], F') → f_5
  149. Rotate(c_5, If[9 < o ≤ 10, ((o - 9) * π / 2), π / 2], K') → g_5
  150. Rotate(e_5, If[9 < o ≤ 10, ((o - 9) * π / 2), π / 2], I') → h_5
  151. Rotate(d_5, If[9 < o ≤ 10, ((o - 9) * π / 2), π / 2], G') → k_5
  152. Rotate(p_2, If[10 < o ≤ 11, ((o - 10) * (π / 2 + υ)), π / 2 + υ], C) → p_5
  153. Rotate(e_4, If[10 < o ≤ 11, ((o - 10) * (π / 2 + σ)), π / 2 + σ], A) → q_5
  154. Rotate(d_4, If[10 < o ≤ 11, ((o - 10) * (π / 2 + τ)), π / 2 + τ], B) → r_5
  155. Dilate(C, If[11 < o ≤ 12, 12 - o, 0], M) → C'_7
  156. Dilate(B, If[11 < o ≤ 12, 12 - o, 0], P) → B'_3
  157. Dilate(A, If[11 < o ≤ 12, 12 - o, 0], O) → A'_3
  158. Dilate(D, If[11 < o ≤ 12, 12 - o, 0], N) → D'_4
  159. Segment(C, C'_7) → i_4
  160. Segment(D, D'_4) → j_4
  161. Segment(A, A'_3) → l_4
  162. Segment(B, B'_3) → m_4
  163. Rotate(C, If[12 < o ≤ 13, ((o - 12) * π), π], M) → C'_8
  164. Rotate(C'_7, If[12 < o ≤ 13, ((o - 12) * π), π], M) → S_1
  165. Segment(C'_8, S_1) → n_4
  166. Rotate(B, If[12 < o ≤ 13, ((o - 12) * π), π], P) → B'_4
  167. Rotate(B'_3, If[12 < o ≤ 13, ((o - 12) * π), π], P) → T_1
  168. Segment(B'_4, T_1) → a_4
  169. Rotate(A, If[12 < o ≤ 13, ((o - 12) * π), π], O) → A'_4
  170. Rotate(A'_3, If[12 < o ≤ 13, ((o - 12) * π), π], O) → U_1
  171. Segment(A'_4, U_1) → b_4
  172. Rotate(D, If[12 < o ≤ 13, ((o - 12) * π), π], N) → D'_5
  173. Rotate(D'_4, If[12 < o ≤ 13, ((o - 12) * π), π], N) → V_1
  174. Segment(D'_5, V_1) → i_5
  175. Dilate(M, If[13 < o ≤ 14, 14 - o, 0], O) → M'
  176. Segment(M, M') → h_2
  177. Rotate(M, If[14 < o ≤ 15, ((-(o - 14)) * π / 2), ((-π)) / 2], O) → M'_1
  178. Rotate(M', If[14 < o ≤ 15, ((-(o - 14)) * π / 2), ((-π)) / 2], O) → M''
  179. Segment(M'_1, M'') → j_5
  180. Dilate(O, If[15 < o ≤ 16, 16 - o, 0], N) → O'
  181. Vector(O, O') → u_4
  182. Translate(M'_1, u_4) → W_1
  183. Translate(M'', u_4) → M'''
  184. Segment(W_1, M''') → l_5
  185. Dilate(I, If[16 < o ≤ 17, 17 - o, 0], U) → I'_1
  186. Vector(I, I'_1) → v_4
  187. Translate(r_4, v_4) → s_5
  188. Line(I, h_2) → m_5
  189. Line(I, l_5) → n_5
  190. Point(m_5) → Z_1
  191. Angle(A, I, Z_1) → ω
  192. Rotate(s_5, If[16 < o ≤ 17, ((o - 16) * ω), ω], I'_1) → t_5
  193. Rotate(t_5, If[18 < o ≤ 19, ((o - 18) * π / 2), π / 2], U) → c_6