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平行四边形与正方形动画演示

平行四边形与正方形动画演示
几何画板及绘图内容加载中，请稍等……
2016-06-29 09:05
多边形向量解析几何变换几何作图
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
①本范例通过动画演示平行四边形与正方形的几何性质与变换关系；②涉及核心数学概念包括向量、几何变换、坐标几何、几何作图等；③构造过程中运用了Segment、Vector、Line、Slope、PointIn、Circle、Dilate、Translate、OrthogonalLine、Intersect、Rotate、Angle、Min、CircleSector、Polygon、Midpoint、Point等GeoGebra命令与对象；④适合高中数学教学，用于理解几何变换与性质，提升空间想象能力。

GeoGebra 命令及参数

  1. Segment(A, B) → h
  2. Vector(A, B) → u
  3. Line(A, B) → a
  4. Slope(a) → m
  5. PointIn(c) → C
  6. Circle(C, h) → p
  7. Dilate(A, If[0 < k ≤ 1, 1 - k, 0], C) → A'
  8. Vector(A, A') → v
  9. Translate(B, v) → B'
  10. Translate(A, v) → A'_1
  11. Segment(A'_1, B') → h'
  12. Segment(A, C) → d
  13. OrthogonalLine(C, d) → g_1
  14. OrthogonalLine(A, d) → f_1
  15. Circle(A, d) → t
  16. Intersect(t, f_1, 2) → H
  17. OrthogonalLine(H, g_1) → h_1
  18. Intersect(g_1, h_1) → I
  19. Translate(C, u) → C'
  20. Dilate(C', If[5 < k ≤ 6, 6 - k, 0], C) → C''_1
  21. Vector(C', C''_1) → v_1
  22. Rotate(B', If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], C') → B''
  23. Rotate(A'_1, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], C') → N
  24. Segment(N, B'') → h''
  25. Segment(C, C') → n
  26. OrthogonalLine(C, n) → q
  27. Intersect(p, q, 2) → F
  28. Angle(F, C, I) → γ
  29. Dilate(C, If[1 < k ≤ 2, 2 - k, 0], C') → C'_1
  30. Vector(C, C'_1) → w
  31. Translate(C, w) → C'_2
  32. Translate(A, w) → A'_2
  33. Segment(A'_2, C'_2) → d'
  34. Min(h', d') → o
  35. Circle(C', (0.15 * o)) → e_1
  36. Intersect(e_1, h', 1) → P
  37. Intersect(e_1, h'', 1) → O
  38. CircleSector(C', O, P) → s_1
  39. Translate(s_1, v_1) → r_2
  40. Line(C, a) → e
  41. Polygon(A, B, C') → poly1, c', a_1, b_1
  42. Polygon(A, C', C) → poly2, c_1, a_2, c'_1
  43. Polygon(A, B, C', C) → poly3, a_3, b_2, c'_2, c_2
  44. Circle(B, d) → f
  45. Segment(B, C') → g
  46. OrthogonalLine(B, g) → i
  47. OrthogonalLine(C', g) → j
  48. Intersect(f, i, 1) → D
  49. OrthogonalLine(D, i) → l
  50. Intersect(j, l) → E
  51. OrthogonalLine(C', n) → r
  52. OrthogonalLine(F, q) → s
  53. Intersect(r, s) → G
  54. Circle(A, h) → d_1
  55. OrthogonalLine(A, h) → i_1
  56. Intersect(d_1, i_1, 1) → J
  57. OrthogonalLine(J, i_1) → j_1
  58. OrthogonalLine(B, h) → k_1
  59. Intersect(j_1, k_1) → K
  60. Rotate(A'_2, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], B) → L
  61. Rotate(C'_2, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], B) → M
  62. Segment(L, M) → d''
  63. Circle(B, (0.25 * o)) → p_1
  64. Circle(C, (0.25 * o)) → q_1
  65. Circle(A, (0.25 * o)) → r_1
  66. Rotate(A, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], C) → A'_3
  67. Rotate(C, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], C) → C'_3
  68. Segment(A'_3, C'_3) → d'_1
  69. Rotate(A, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], A) → A'_4
  70. Rotate(B, If[2 < k ≤ 3, ((-(k - 2)) * π / 2), ((-π)) / 2], A) → B'_1
  71. Segment(A'_4, B'_1) → h'_1
  72. Rotate(N, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], G) → N'
  73. Rotate(B'', If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], G) → B'''
  74. Segment(N', B''') → h'''
  75. Rotate(A'_3, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], I) → Q
  76. Rotate(C'_3, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], I) → R
  77. Segment(Q, R) → l_1
  78. Rotate(A'_4, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], J) → S
  79. Rotate(B'_1, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], J) → T
  80. Segment(S, T) → m_1
  81. Rotate(L, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], D) → L'
  82. Rotate(M, If[3 < k ≤ 4, ((-(k - 3)) * π / 2), ((-π)) / 2], D) → M'
  83. Segment(L', M') → d'''
  84. Rotate(N', If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], F) → N''
  85. Rotate(B''', If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], F) → U
  86. Segment(N'', U) → n_1
  87. Rotate(Q, If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], H) → Q'
  88. Rotate(R, If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], H) → R'
  89. Segment(Q', R') → t_1
  90. Rotate(S, If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], K) → S'
  91. Rotate(T, If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], K) → T'
  92. Segment(S', T') → f_2
  93. Rotate(L', If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], E) → L''
  94. Rotate(M', If[4 < k ≤ 5, ((-(k - 4)) * π / 2), ((-π)) / 2], E) → M''
  95. Segment(L'', M'') → g_2
  96. Polygon(C, C', G, F) → poly4, c_3, c'_3, g_3, f_3
  97. Polygon(A, B, K, J) → poly5, a_4, b_3, k_2, j_2
  98. Polygon(B, D, E, C') → poly6, b_4, d_2, e_2, c'_4
  99. Polygon(A, C, I, H) → poly7, a_5, c_4, i_2, h_2
  100. Dilate(C', If[5 < k ≤ 6, 6 - k, 0], G) → C''
  101. Dilate(C', If[5 < k ≤ 6, 6 - k, 0], A) → C''_2
  102. Dilate(C', If[5 < k ≤ 6, 6 - k, 0], B) → C''_3
  103. Vector(C', C'') → u_1
  104. Vector(C', C''_2) → w_1
  105. Vector(C', C''_3) → u_2
  106. Translate(s_1, u_1) → p_2
  107. Translate(s_1, w_1) → q_2
  108. Translate(s_1, u_2) → s_2
  109. Line(C, C') → l_2
  110. Intersect(l_2, l_1) → V
  111. Rotate(p_2, If[6 < k ≤ 7, ((k - 6) * π / 2), π / 2], G) → d_3
  112. Rotate(q_2, If[6 < k ≤ 7, ((k - 6) * π), π], A) → e_3
  113. Line(B, u_1) → m_2
  114. Intersect(l_2, m_2) → W
  115. Rotate(s_2, If[6 < k ≤ 7, ((k - 6) * (π + γ)), π + γ], B) → h_3
  116. Dilate(G, If[7 < k ≤ 8, 8 - k, 0], F) → G'
  117. Dilate(C, If[7 < k ≤ 8, 8 - k, 0], I) → C'_4
  118. Dilate(A, If[7 < k ≤ 8, 8 - k, 0], J) → A'_5
  119. Dilate(B, If[7 < k ≤ 8, 8 - k, 0], D) → B'_2
  120. Vector(G, G') → v_2
  121. Vector(C, C'_4) → w_2
  122. Vector(A, A'_5) → n_2
  123. Vector(B, B'_2) → u_3
  124. Translate(d_3, v_2) → k_3
  125. Rotate(r_2, If[6 < k ≤ 7, ((k - 6) * γ), γ], C) → t_2
  126. Translate(t_2, w_2) → p_3
  127. Translate(e_3, n_2) → q_3
  128. Translate(h_3, u_3) → r_3
  129. Rotate(k_3, If[8 < k ≤ 9, ((k - 8) * π / 2), π / 2], F) → s_3
  130. Rotate(p_3, If[8 < k ≤ 9, ((k - 8) * π / 2), π / 2], I) → t_3
  131. Rotate(r_3, If[8 < k ≤ 9, ((k - 8) * π / 2), π / 2], D) → d_4
  132. Rotate(q_3, If[8 < k ≤ 9, ((k - 8) * π / 2), π / 2], J) → e_4
  133. Dilate(F, If[9 < k ≤ 10, 10 - k, 0], C) → F'
  134. Dilate(I, If[9 < k ≤ 10, 10 - k, 0], H) → I'
  135. Dilate(J, If[9 < k ≤ 10, 10 - k, 0], K) → J'
  136. Dilate(D, If[9 < k ≤ 10, 10 - k, 0], E) → D'
  137. Vector(D, D') → v_3
  138. Vector(J, J') → w_3
  139. Vector(I, I') → i_3
  140. Vector(F, F') → j_3
  141. Translate(t_3, i_3) → f_4
  142. Translate(s_3, j_3) → g_4
  143. Translate(d_4, v_3) → h_4
  144. Translate(e_4, w_3) → k_4
  145. Rotate(h_4, If[10 < k ≤ 11, ((k - 10) * π / 2), π / 2], D') → p_4
  146. Rotate(f_4, If[10 < k ≤ 11, ((k - 10) * π / 2), π / 2], H) → q_4
  147. Rotate(g_4, If[10 < k ≤ 11, ((k - 10) * π / 2), π / 2], C) → r_4
  148. Rotate(k_4, If[10 < k ≤ 11, ((k - 10) * π / 2), π / 2], K) → s_4
  149. Angle(C, C', B) → α
  150. Rotate(s_1, If[11 < k ≤ 12, ((k - 11) * (π / 2 + α)), π / 2 + α], C') → t_4
  151. Rotate(e_3, If[11 < k ≤ 12, ((k - 11) * (α + π / 2)), α + π / 2], A) → c_5
  152. Rotate(h_3, If[11 < k ≤ 12, ((k - 11) * ((3 * π / 2) - α)), (3 * π / 2) - α], B) → d_5
  153. Segment(B, E) → m_3
  154. Segment(A, K) → n_3
  155. Segment(C, H) → i_4
  156. Midpoint(i_4) → B_1
  157. Midpoint(n_3) → C_1
  158. Midpoint(m_3) → D_1
  159. Segment(C', F) → l_3
  160. Midpoint(l_3) → Z
  161. Dilate(C, If[12 < k ≤ 13, 13 - k, 0], Z) → C''_4
  162. Dilate(A, If[12 < k ≤ 13, 13 - k, 0], B_1) → C'_5
  163. Dilate(B, If[12 < k ≤ 13, 13 - k, 0], C_1) → A'_6
  164. Dilate(C', If[12 < k ≤ 13, 13 - k, 0], D_1) → B'_3
  165. Point(h'_1) → E_1
  166. Segment(B, A'_6) → m_4
  167. Segment(C', B'_3) → n_4
  168. Segment(C, C''_4) → j_4
  169. Dilate(Z, If[13 < k ≤ 14, 14 - k, 0], G) → Z'
  170. Dilate(B_1, If[13 < k ≤ 14, 14 - k, 0], I) → A_1
  171. Dilate(C_1, If[13 < k ≤ 14, 14 - k, 0], J) → F_1
  172. Dilate(D_1, If[13 < k ≤ 14, 14 - k, 0], D) → G_1
  173. Segment(Z, Z') → f_5
  174. Segment(D_1, G_1) → g_5
  175. Segment(B_1, A_1) → h_5
  176. Segment(C_1, F_1) → i_5
  177. Rotate(C, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], Z) → C'_6
  178. Rotate(C''_4, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], Z) → H_1
  179. Segment(C'_6, H_1) → j_5
  180. Rotate(A, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], B_1) → I_1
  181. Rotate(B_1, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], B_1) → J_1
  182. Segment(I_1, J_1) → k_5
  183. Rotate(B, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], C_1) → B'_4
  184. Rotate(A'_6, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], C_1) → K_1
  185. Segment(B'_4, K_1) → l_5
  186. Rotate(C', If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], D_1) → C''_5
  187. Rotate(B'_3, If[14 < k ≤ 15, ((-(k - 14)) * π), (-π)], D_1) → L_1
  188. Segment(C''_5, L_1) → m_5
  189. Dilate(D_1, If[15 < k ≤ 16, 16 - k, 0], Z) → M_1
  190. Segment(D_1, M_1) → n_5
  191. Rotate(D_1, If[16 < k ≤ 17, ((-(k - 16)) * π), (-π)], Z) → N_1
  192. Rotate(M_1, If[16 < k ≤ 17, ((-(k - 16)) * π), (-π)], Z) → O_1
  193. Rotate(D_1, If[16 < k ≤ 17, ((-(k - 16)) * π / 2), ((-π)) / 2], Z) → P_1
  194. Rotate(M_1, If[16 < k ≤ 17, ((-(k - 16)) * π / 2), ((-π)) / 2], Z) → Q_1
  195. Segment(P_1, Q_1) → p_5
  196. Rotate(P_1, If[17 < k ≤ 18, ((-(k - 17)) * π / 2), ((-π)) / 2], B_1) → R_1
  197. Rotate(Q_1, If[17 < k ≤ 18, ((-(k - 17)) * π / 2), ((-π)) / 2], B_1) → S_1
  198. Segment(R_1, S_1) → q_5
  199. Rotate(R_1, If[18 < k ≤ 19, ((-(k - 18)) * π / 2), ((-π)) / 2], C_1) → T_1
  200. Rotate(S_1, If[18 < k ≤ 19, ((-(k - 18)) * π / 2), ((-π)) / 2], C_1) → U_1
  201. Segment(T_1, U_1) → r_5
  202. Polygon(D_1, Z, B_1, C_1) → poly8, d_6, z_1, b_5, c_6
  203. Dilate(E, If[19 < k ≤ 20, 20 - k, 0], D_1) → E'
  204. Dilate(F, If[19 < k ≤ 20, 20 - k, 0], Z) → F'_1
  205. Dilate(H, If[19 < k ≤ 20, 20 - k, 0], B_1) → H'
  206. Dilate(K, If[19 < k ≤ 20, 20 - k, 0], C_1) → K'
  207. Vector(K, K') → u_4
  208. Vector(H, H') → v_4
  209. Vector(E, E') → w_4
  210. Vector(F, F'_1) → u_5
  211. Translate(s_3, u_5) → e_5
  212. Translate(q_4, v_4) → s_5
  213. Translate(s_4, u_4) → t_5
  214. Translate(p_4, w_4) → e_6
  215. Line(D_1, g_2) → f_6
  216. Point(f_6) → W_1
  217. Angle(Z, D_1, W_1) → β
  218. Rotate(e_6, If[20 < k ≤ 21, ((-(k - 20)) * β), (-β)], D_1) → g_6
  219. Rotate(s_5, If[20 < k ≤ 21, ((-(k - 20)) * β), (-β)], B_1) → h_6
  220. Line(C_1, f_2) → i_6
  221. Point(i_6) → Z_1
  222. Angle(D_1, C_1, Z_1) → δ
  223. Rotate(t_5, If[20 < k ≤ 21, ((-(k - 20)) * δ), (-δ)], C_1) → k_6
  224. Rotate(e_5, If[20 < k ≤ 21, ((-(k - 20)) * δ), (-δ)], Z) → p_6
  225. Segment(A, C'_5) → l_4