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任意三角形三外接正方形的三个结果现象

任意三角形三外接正方形的三个结果现象
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2017-04-28 05:37
测试改名0变换
本范例展示了在任意三角形的三边外分别作正方形，由此产生的三个有趣现象。核心数学概念包括三角形、正方形和几何变换。通过使用GeoGebra命令如Circle、Polygon、Dilate等进行构造。适合初中及以上学生或教师教学使用。

GeoGebra 命令及参数

  1. Circle(A, b) → d
  2. Point(d) → B
  3. Circle(B, c) → e
  4. Point(e) → C
  5. Polygon(A, B, 4) → poly1, f, g, h, i, D, E
  6. Polygon(C, B, 4) → poly2, j, k, l, m, F, G
  7. Midpoint(D, A) → H
  8. Midpoint(F, C) → I
  9. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], A) → B'
  10. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], F) → B'_1
  11. Segment(B, B') → n
  12. Segment(B, B'_1) → p
  13. Rotate(B, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → B'_2
  14. Rotate(B'_1, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → L
  15. Segment(B'_2, L) → p'
  16. Rotate(B, If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B'_3
  17. Rotate(B', If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B''
  18. Segment(B'_3, B'') → n'
  19. Rotate(B'_2, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → M
  20. Rotate(L, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → L'
  21. Segment(M, L') → p''
  22. Rotate(B'_3, If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → N
  23. Rotate(B'', If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → B'''
  24. Segment(N, B''') → n''
  25. Rotate(N, If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → N'
  26. Rotate(B''', If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → O
  27. Segment(N', O) → n'''
  28. Rotate(M, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → M'
  29. Rotate(L', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → L''
  30. Segment(M', L'') → p'''
  31. Segment(F, G) → q
  32. Segment(F, B) → r
  33. Segment(A, B) → s
  34. Min(r, s) → o
  35. Circle(F, (0.15 * o)) → t
  36. Intersect(l, q) → P
  37. Intersect(t, q) → Q,
  38. Intersect(t, r) → R,
  39. CircleSector(F, Q, R) → c_1
  40. Dilate(F, If[4 < a ≤ 5, 5 - a, 0], C) → F'
  41. Vector(F, F') → u
  42. Translate(c_1, u) → d_1
  43. Rotate(d_1, If[5 < a ≤ 6, ((a - 5) * π), π], C) → e_1
  44. Dilate(F, If[6 < a ≤ 7, 7 - a, 0], A) → F'_1
  45. Dilate(C, If[6 < a ≤ 7, 7 - a, 0], E) → C'
  46. Vector(F, F'_1) → v
  47. Vector(C, C') → w
  48. Translate(e_1, w) → f_1
  49. Translate(c_1, v) → g_1
  50. Line(A, p) → h_1
  51. Line(E, p''') → i_1
  52. Point(h_1) → S
  53. Angle(S, A, E) → α
  54. Rotate(g_1, If[7 < a ≤ 8, ((-(a - 7)) * ((2 * π) - α)), (-((2 * π) - α))], A) → k_1
  55. Line(E, h_1) → j_1
  56. Point(j_1) → T
  57. Angle(D, E, T) → β
  58. Rotate(f_1, If[7 < a ≤ 8, ((-(a - 7)) * β), (-β)], E) → p_1
  59. Dilate(E, If[8 < a ≤ 9, 9 - a, 0], G) → E'
  60. Dilate(F, If[8 < a ≤ 9, 9 - a, 0], D) → F'_2
  61. Vector(E, E') → u_1
  62. Vector(F, F'_2) → v_1
  63. Translate(p_1, u_1) → q_1
  64. Translate(c_1, v_1) → r_1
  65. Line(G, n'') → l_1
  66. Point(l_1) → U
  67. Line(G, n') → m_1
  68. Point(m_1) → V
  69. Angle(C, G, V) → γ
  70. Rotate(q_1, If[8 < a ≤ 9, ((-(a - 8)) * γ), (-γ)], E') → s_1
  71. Line(D, p) → n_1
  72. Point(n_1) → W
  73. Angle(W, D, B) → δ
  74. Rotate(r_1, If[8 < a ≤ 9, ((a - 8) * δ), δ], F'_2) → t_1
  75. Midpoint(A, B) → Z
  76. Midpoint(B, C) → A_1
  77. Midpoint(Z, B) → B_1
  78. Midpoint(A_1, B) → C_1
  79. Dilate(C, If[9 < a ≤ 10, 10 - a, 0], C_1) → C'_1
  80. Dilate(A, If[9 < a ≤ 10, 10 - a, 0], B_1) → A'
  81. Vector(A, A') → w_1
  82. Vector(C, C'_1) → a_1
  83. Translate(e_1, a_1) → c_2
  84. Translate(k_1, w_1) → d_2
  85. Rotate(d_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], B_1) → e_2
  86. Rotate(c_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], C_1) → f_2
  87. Dilate(C_1, If[11 < a ≤ 12, 12 - a, 0], B) → D_1
  88. Dilate(B_1, If[11 < a ≤ 12, 12 - a, 0], B) → E_1
  89. Vector(B_1, E_1) → b_1
  90. Vector(C_1, D_1) → u_2
  91. Translate(e_2, b_1) → g_2
  92. Translate(f_2, u_2) → h_2
  93. Dilate(F, If[12 < a ≤ 13, 13 - a, 0], I) → F'_3
  94. Segment(F, F'_3) → i_2
  95. Dilate(D, If[12 < a ≤ 13, 13 - a, 0], H) → D'
  96. Segment(D, D') → j_2
  97. Rotate(F, If[13 < a ≤ 14, ((a - 13) * π), π], I) → F'_4
  98. Rotate(F'_3, If[13 < a ≤ 14, ((a - 13) * π), π], I) → F_1
  99. Segment(F'_4, F_1) → k_2
  100. Rotate(D, If[13 < a ≤ 14, ((a - 13) * π), π], H) → D'_1
  101. Rotate(D', If[13 < a ≤ 14, ((a - 13) * π), π], H) → D''
  102. Segment(D'_1, D'') → l_2
  103. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], I) → B'_4
  104. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], H) → B'_5
  105. Vector(B, B'_4) → v_2
  106. Vector(B, B'_5) → w_2
  107. Line(I, p) → g_3
  108. Point(g_3) → N_1
  109. Line(H, n''') → l_3
  110. Point(l_3) → O_1
  111. Polygon(A, F, B) → poly3, b_2, a_2, f_3
  112. Circle(A, 2) → c_4
  113. Circle(F, 2) → d_4
  114. Polygon(F, A, 4) → poly4, i_4, j_4, k_4, l_4, C_2, D_2
  115. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], C_2) → A'_1
  116. Segment(A, A'_1) → m_2
  117. Rotate(A, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], C_2) → A'_2
  118. Rotate(A'_1, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], C_2) → J
  119. Segment(A'_2, J) → n_2
  120. Rotate(A'_2, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], D_2) → K
  121. Rotate(J, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], D_2) → J'
  122. Segment(K, J') → p_2
  123. Rotate(K, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], F) → K'
  124. Rotate(J', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], F) → J''
  125. Segment(K', J'') → q_2
  126. Angle(B, F, D_2) → ε
  127. Rotate(c_1, If[12 < a ≤ 13, ((a - 12) * ε), ε], F) → r_2
  128. Midpoint(A, D_2) → G_1
  129. Dilate(A, If[12 < a ≤ 13, 13 - a, 0], G_1) → A'_3
  130. Segment(A, A'_3) → s_2
  131. Rotate(A, If[13 < a ≤ 14, ((a - 13) * π), π], G_1) → A'_4
  132. Rotate(A'_3, If[13 < a ≤ 14, ((a - 13) * π), π], G_1) → H_1
  133. Segment(A'_4, H_1) → t_2
  134. Line(G_1, B) → h_3
  135. Line(A, I) → i_3
  136. Line(F, H) → j_3
  137. Intersect(i_3, j_3) → I_1
  138. Line(H, I) → k_3
  139. Distance(H, I) → distanceHI
  140. Distance(B, G_1) → distanceBG1
  141. Dilate(G_1, If[14 < a ≤ 15, 15 - a, 0], B) → J_1
  142. Segment(G_1, J_1) → m_3
  143. Dilate(A, If[15 < a ≤ 16, 16 - a, 0], I) → A'_5
  144. Segment(A, A'_5) → n_3
  145. Dilate(F, If[16 < a ≤ 17, 17 - a, 0], I_1) → F'_5
  146. Segment(F, F'_5) → p_3
  147. Dilate(I_1, If[17 < a ≤ 18, 18 - a, 0], H) → K_1
  148. Segment(I_1, K_1) → q_3
  149. Line(H, I) → r_3
  150. Line(B, G_1) → s_3
  151. Intersect(r_3, s_3) → L_1
  152. OrthogonalLine(C, r_3) → t_3
  153. Intersect(t_3, r_3) → M_1
  154. Line(H, I) → a_3
  155. Line(G_1, a_3) → b_3
  156. Rotate(G_1, If[18 < a ≤ 19, ((-(a - 18)) * π / 2), ((-π)) / 2], G_1) → P_1
  157. Rotate(J_1, If[18 < a ≤ 19, ((-(a - 18)) * π / 2), ((-π)) / 2], G_1) → Q_1
  158. Segment(P_1, Q_1) → c_3
  159. Dilate(G_1, If[19 < a ≤ 20, 20 - a, 0], H) → R_1
  160. Vector(G_1, R_1) → u_3
  161. Translate(P_1, u_3) → S_1
  162. Translate(Q_1, u_3) → T_1
  163. Segment(S_1, T_1) → d_3
  164. Dilate(C, If[20 < a ≤ 21, 21 - a, 0], M_1) → C'_2
  165. Vector(C, C'_2) → v_3
  166. Translate(e_1, v_3) → e_3
  167. Line(C, d_3) → f_4
  168. Point(f_4) → U_1
  169. Angle(G, C, U_1) → ζ
  170. Rotate(e_3, If[21 < a ≤ 22, ((a - 21) * ζ), ζ], M_1) → e_4
  171. Dilate(M_1, If[22 < a ≤ 23, 23 - a, 0], L_1) → V_1
  172. Vector(M_1, V_1) → w_3
  173. Translate(e_4, w_3) → g_4
  174. Line(I_1, B) → h_4
  175. Circle(B, 1) → p_4
  176. Intersect(p_4, h_4, 2) → W_1
  177. Ray(I_1, B) → m_4
  178. Circle(G_1, 2) → q_4
  179. Intersect(q_4, s_3, 2) → Z_1
  180. Ray(G_1, Z_1) → n_4