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三角形三边外侧构建正方形时的动态现象

三角形三边外侧构建正方形时的动态现象
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2017-04-30 08:48
测试改名0多边形向量几何作图
①本范例演示当在任意三角形的每条边上构建正方形时，动态产生的各种几何现象；②涉及的核心数学概念包括几何变换、向量、角度、多边形构造等；③作图要点提炼自构造摘要，使用了Circle（圆）、Point（点）、Polygon（多边形）、Midpoint（中点）、Segment（线段）、Dilate（缩放）、Rotate（旋转）、Min（最小值）、Intersect（交点）、CircleSector（扇形）、Vector（向量）、Translate（平移）、Line（直线）、Angle（角度）、Distance（距离）、OrthogonalLine（垂线）等GeoGebra命令和对象；④适合高中及以上数学学习者，用于探索几何变换和动态几何中的规律。

GeoGebra 命令及参数

  1. Circle(A, b) → d
  2. Point(d) → B
  3. Polygon(A, B, 4) → poly1, f, g, h, i, D, E
  4. Circle(B, c) → e
  5. Point(e) → C
  6. Polygon(C, B, 4) → poly2, j, k, l, m, F, G
  7. Midpoint(E, G) → A_2
  8. Segment(A, A_2) → b_4
  9. Midpoint(D, A) → H
  10. Midpoint(F, C) → I
  11. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], A) → B'
  12. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], F) → B'_1
  13. Segment(B, B') → n
  14. Segment(B, B'_1) → p
  15. Rotate(B, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → B'_2
  16. Rotate(B'_1, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → L
  17. Segment(B'_2, L) → p'
  18. Rotate(B, If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B'_3
  19. Rotate(B', If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B''
  20. Segment(B'_3, B'') → n'
  21. Rotate(B'_2, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → M
  22. Rotate(L, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → L'
  23. Segment(M, L') → p''
  24. Rotate(B'_3, If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → N
  25. Rotate(B'', If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → B'''
  26. Segment(N, B''') → n''
  27. Rotate(N, If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → N'
  28. Rotate(B''', If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → O
  29. Segment(N', O) → n'''
  30. Rotate(M, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → M'
  31. Rotate(L', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → L''
  32. Segment(M', L'') → p'''
  33. Segment(F, G) → q
  34. Segment(F, B) → r
  35. Segment(A, B) → s
  36. Min(r, s) → o
  37. Intersect(l, q) → P
  38. Circle(F, (0.3 * b_4)) → t
  39. Intersect(t, q) → Q,
  40. Intersect(t, r) → R,
  41. CircleSector(F, Q, R) → c_1
  42. Dilate(F, If[4 < a ≤ 5, 5 - a, 0], C) → F'
  43. Vector(F, F') → u
  44. Translate(c_1, u) → d_1
  45. Rotate(d_1, If[5 < a ≤ 6, ((a - 5) * π), π], C) → e_1
  46. Dilate(F, If[6 < a ≤ 7, 7 - a, 0], A) → F'_1
  47. Dilate(C, If[6 < a ≤ 7, 7 - a, 0], E) → C'
  48. Vector(F, F'_1) → v
  49. Vector(C, C') → w
  50. Translate(e_1, w) → f_1
  51. Translate(c_1, v) → g_1
  52. Line(A, p) → h_1
  53. Line(E, p''') → i_1
  54. Point(h_1) → S
  55. Angle(S, A, E) → α
  56. Rotate(g_1, If[7 < a ≤ 8, ((-(a - 7)) * ((2 * π) - α)), (-((2 * π) - α))], A) → k_1
  57. Line(E, h_1) → j_1
  58. Point(j_1) → T
  59. Angle(D, E, T) → β
  60. Rotate(f_1, If[7 < a ≤ 8, ((-(a - 7)) * β), (-β)], E) → p_1
  61. Dilate(E, If[8 < a ≤ 9, 9 - a, 0], G) → E'
  62. Dilate(F, If[8 < a ≤ 9, 9 - a, 0], D) → F'_2
  63. Vector(E, E') → u_1
  64. Vector(F, F'_2) → v_1
  65. Translate(p_1, u_1) → q_1
  66. Translate(c_1, v_1) → r_1
  67. Line(G, n'') → l_1
  68. Point(l_1) → U
  69. Line(G, n') → m_1
  70. Point(m_1) → V
  71. Angle(C, G, V) → γ
  72. Rotate(q_1, If[8 < a ≤ 9, ((-(a - 8)) * γ), (-γ)], E') → s_1
  73. Line(D, p) → n_1
  74. Point(n_1) → W
  75. Angle(W, D, B) → δ
  76. Rotate(r_1, If[8 < a ≤ 9, ((a - 8) * δ), δ], F'_2) → t_1
  77. Midpoint(A, B) → Z
  78. Midpoint(B, C) → A_1
  79. Midpoint(Z, B) → B_1
  80. Midpoint(A_1, B) → C_1
  81. Dilate(C, If[9 < a ≤ 10, 10 - a, 0], C_1) → C'_1
  82. Dilate(A, If[9 < a ≤ 10, 10 - a, 0], B_1) → A'
  83. Vector(A, A') → w_1
  84. Vector(C, C'_1) → a_1
  85. Translate(e_1, a_1) → c_2
  86. Translate(k_1, w_1) → d_2
  87. Rotate(d_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], B_1) → e_2
  88. Rotate(c_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], C_1) → f_2
  89. Dilate(C_1, If[11 < a ≤ 12, 12 - a, 0], B) → D_1
  90. Dilate(B_1, If[11 < a ≤ 12, 12 - a, 0], B) → E_1
  91. Vector(B_1, E_1) → b_1
  92. Vector(C_1, D_1) → u_2
  93. Translate(e_2, b_1) → g_2
  94. Translate(f_2, u_2) → h_2
  95. Dilate(F, If[12 < a ≤ 13, 13 - a, 0], I) → F'_3
  96. Segment(F, F'_3) → i_2
  97. Dilate(D, If[12 < a ≤ 13, 13 - a, 0], H) → D'
  98. Segment(D, D') → j_2
  99. Rotate(F, If[13 < a ≤ 14, ((a - 13) * π), π], I) → F'_4
  100. Rotate(F'_3, If[13 < a ≤ 14, ((a - 13) * π), π], I) → F_1
  101. Segment(F'_4, F_1) → k_2
  102. Rotate(D, If[13 < a ≤ 14, ((a - 13) * π), π], H) → D'_1
  103. Rotate(D', If[13 < a ≤ 14, ((a - 13) * π), π], H) → D''
  104. Segment(D'_1, D'') → l_2
  105. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], I) → B'_4
  106. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], H) → B'_5
  107. Vector(B, B'_4) → v_2
  108. Vector(B, B'_5) → w_2
  109. Line(I, p) → g_3
  110. Point(g_3) → N_1
  111. Line(H, n''') → l_3
  112. Point(l_3) → O_1
  113. Polygon(A, F, B) → poly3, b_2, a_2, f_3
  114. Circle(A, 2) → c_4
  115. Circle(F, 2) → d_4
  116. Polygon(F, A, 4) → poly4, i_4, j_4, k_4, l_4, C_2, D_2
  117. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], C_2) → A'_1
  118. Segment(A, A'_1) → m_2
  119. Rotate(A, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], C_2) → A'_2
  120. Rotate(A'_1, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], C_2) → J
  121. Segment(A'_2, J) → n_2
  122. Rotate(A'_2, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], D_2) → K
  123. Rotate(J, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], D_2) → J'
  124. Segment(K, J') → p_2
  125. Rotate(K, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], F) → K'
  126. Rotate(J', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], F) → J''
  127. Segment(K', J'') → q_2
  128. Angle(B, F, D_2) → ε
  129. Rotate(c_1, If[12 < a ≤ 13, ((a - 12) * ε), ε], F) → r_2
  130. Midpoint(A, D_2) → G_1
  131. Dilate(A, If[12 < a ≤ 13, 13 - a, 0], G_1) → A'_3
  132. Segment(A, A'_3) → s_2
  133. Rotate(A, If[13 < a ≤ 14, ((a - 13) * π), π], G_1) → A'_4
  134. Rotate(A'_3, If[13 < a ≤ 14, ((a - 13) * π), π], G_1) → H_1
  135. Segment(A'_4, H_1) → t_2
  136. Line(G_1, B) → h_3
  137. Line(A, I) → i_3
  138. Line(F, H) → j_3
  139. Line(H, I) → k_3
  140. Distance(H, I) → distanceHI
  141. Distance(B, G_1) → distanceBG1
  142. Dilate(G_1, If[14 < a ≤ 15, 15 - a, 0], B) → J_1
  143. Dilate(A, If[15 < a ≤ 16, 16 - a, 0], I) → A'_5
  144. Line(H, I) → r_3
  145. Line(B, G_1) → s_3
  146. Intersect(r_3, s_3) → L_1
  147. OrthogonalLine(C, r_3) → t_3
  148. Intersect(t_3, r_3) → M_1
  149. Line(H, I) → a_3
  150. Line(G_1, a_3) → b_3
  151. Rotate(G_1, If[18 < a ≤ 19, ((-(a - 18)) * π / 2), ((-π)) / 2], G_1) → P_1
  152. Dilate(C, If[20 < a ≤ 21, 21 - a, 0], M_1) → C'_2
  153. Vector(C, C'_2) → v_3
  154. Dilate(M_1, If[22 < a ≤ 23, 23 - a, 0], L_1) → V_1
  155. Vector(M_1, V_1) → w_3
  156. Circle(B, 1) → p_4
  157. Circle(G_1, 2) → q_4
  158. Intersect(q_4, s_3, 2) → Z_1
  159. Dilate(E, If[14 < a ≤ 15, 15 - a, 0], A_2) → E'_1
  160. Segment(E, E'_1) → m_3
  161. Dilate(A_2, If[15 < a ≤ 16, 16 - a, 0], G) → B_2
  162. Segment(A_2, B_2) → n_3
  163. Rotate(E, If[16 < a ≤ 17, ((a - 16) * π), π], A_2) → E'_2
  164. Rotate(E'_1, If[16 < a ≤ 17, ((a - 16) * π), π], A_2) → E_2
  165. Segment(E'_2, E_2) → r_4
  166. Dilate(A, If[17 < a ≤ 18, 18 - a, 0], A_2) → A'_6
  167. Segment(A, A'_6) → p_3
  168. Rotate(A, If[18 < a ≤ 19, ((a - 18) * π / 2), π / 2], A_2) → A'_7
  169. Rotate(A'_6, If[18 < a ≤ 19, ((a - 18) * π / 2), π / 2], A_2) → I_1
  170. Segment(A'_7, I_1) → q_3
  171. Rotate(A'_7, If[19 < a ≤ 20, ((a - 19) * π / 2), π / 2], F) → K_1
  172. Rotate(I_1, If[19 < a ≤ 20, ((a - 19) * π / 2), π / 2], F) → Q_1
  173. Segment(K_1, Q_1) → c_3
  174. Rotate(K_1, If[20 < a ≤ 21, ((a - 20) * π / 2), π / 2], G_1) → R_1
  175. Rotate(Q_1, If[20 < a ≤ 21, ((a - 20) * π / 2), π / 2], G_1) → S_1
  176. Segment(R_1, S_1) → d_3
  177. Polygon(A, A_2, F, G_1) → poly5, a_4, a_5, f_4, g_4
  178. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], A_2) → B'_6
  179. Vector(B, B'_6) → u_3
  180. Translate(g_2, u_3) → e_3
  181. Line(B, q_3) → h_4
  182. Point(h_4) → T_1
  183. Angle(A, B, T_1) → ζ
  184. Rotate(e_3, If[21 < a ≤ 22, ((a - 21) * ζ), ζ], B'_6) → e_4
  185. Dilate(D, If[22 < a ≤ 23, 23 - a, 0], A_2) → D'_2
  186. Segment(D, D'_2) → m_4
  187. Rotate(D, If[23 < a ≤ 24, ((-(a - 23)) * π / 2), ((-π)) / 2], A_2) → D'_3
  188. Rotate(D'_2, If[23 < a ≤ 24, ((-(a - 23)) * π / 2), ((-π)) / 2], A_2) → U_1
  189. Segment(D'_3, U_1) → n_4
  190. Dilate(B, If[25 < a ≤ 26, 26 - a, 0], A_2) → B'_7
  191. Vector(B, B'_7) → u_4
  192. Line(B, m_4) → t_4
  193. Point(t_4) → W_1
  194. Angle(C, B, W_1) → κ
  195. Rotate(h_2, If[24 < a ≤ 25, ((a - 24) * κ), κ], B) → s_4
  196. Translate(s_4, u_4) → c_5