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动态正方形中线问题演示

动态正方形中线问题演示
几何画板及绘图内容加载中，请稍等……
2018-03-14 03:55
多边形向量解析几何变换几何作图
①本范例动态演示一个涉及正方形及其两腰中点的几何问题；②核心数学概念包括正方形性质、中点、旋转、缩放、多边形构造等；③作图要点运用了 Rotate（旋转）、Dilate（缩放）、Polygon（多边形）、Midpoint（中点）、CircleArc（圆弧）、Point（点）、Segment（线段）、Intersect（交点）、Circle（圆）、CircleSector（圆扇形）、Vector（向量）、Translate（平移）、Line（直线）、Angle（角度）、Mirror（镜像）、Ray（射线）等 GeoGebra 命令及对象；④适合高中数学教学，用于理解几何变换与动态几何关系。

GeoGebra 命令及参数

  1. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  2. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  3. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  4. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  5. Midpoint(A, C) → V
  6. CircleArc(V, C, D) → g_2
  7. Point(g_2) → B
  8. Segment(C, B) → j_1
  9. Segment(A, B) → n
  10. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  11. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  12. Segment(R, A''') → f''
  13. Intersect(n, f'') → Z
  14. Segment(B, Z) → l_1
  15. Segment(A, A') → f
  16. Segment(A'_1, A'') → f'
  17. Segment(A, B_2) → g
  18. Segment(B_2, C) → h
  19. Segment(C, D) → i
  20. Segment(D, A) → j
  21. Circle(B_2, (0.15 * f)) → p
  22. Circle(C, Radius[p]) → k
  23. Circle(A, (0.1 * j)) → q
  24. Circle(D, (0.1 * j)) → r
  25. Segment(A, C) → l
  26. Segment(D, B_2) → m
  27. Intersect(r, j, 1) → F
  28. Intersect(r, m, 1) → G
  29. Intersect(r, i, 1) → H
  30. Intersect(k, i, 1) → I
  31. Intersect(k, l, 1) → J
  32. Intersect(k, h, 1) → K
  33. Intersect(p, h, 1) → L
  34. Intersect(p, m, 1) → M
  35. Intersect(p, g, 1) → N
  36. Intersect(q, g, 1) → O
  37. Intersect(q, l, 1) → P
  38. Intersect(q, j, 1) → Q
  39. CircleSector(B_2, L, N) → s
  40. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  41. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  42. Segment(R', S) → f'''
  43. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  44. Vector(B_2, B') → u
  45. Translate(s, u) → s'
  46. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  47. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  48. Vector(C, C') → v
  49. Translate(s'', v) → s'''
  50. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  51. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  52. Vector(D, D') → w
  53. Translate(t, w) → t'
  54. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  55. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  56. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  57. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  58. Vector(B_2, B'_1) → v_1
  59. Vector(B_2, B'_3) → w_1
  60. Vector(B_2, B'_2) → i_1
  61. Circle(A, n - j_1) → h_2
  62. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  63. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  64. Vector(B_2, B'_4) → u_1
  65. Translate(s, u_1) → s'_1
  66. Line(B, f') → a_1
  67. Point(a_1) → U
  68. Angle(U, B, C) → α
  69. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  70. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  71. Line(A, B_2) → h_1
  72. Angle(B, A, D) → β
  73. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  74. Mirror(C, D) → C'_1
  75. Rotate(C, ((-π)) / 3, B_2) → C'_2
  76. Segment(B_2, C'_2) → f_1
  77. Segment(C, C'_2) → g_1
  78. Intersect(k, g_1, 1) → E
  79. Intersect(k, h) → T,
  80. CircleSector(C, T, E) → d_1
  81. Rotate(D, 15°, C) → D'_2
  82. Angle(D, C, D'_2) → γ
  83. Ray(C, D'_2) → k_1
  84. Rotate(C, (-(15°)), D) → C'_3
  85. Angle(C'_3, D, C) → δ
  86. Ray(D, C'_3) → m_1
  87. Intersect(k_1, m_1) → W
  88. Rotate(B_2, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → A_1
  89. Rotate(C, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → C'_4
  90. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  91. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  92. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  93. Polygon(C, W, D) → poly2, d_2, c_2, w_2
  94. Polygon(A, W, D) → poly3, d_3, a_2, w_3
  95. Polygon(B_2, W, C) → poly4, c_3, b_2, w_4
  96. Polygon(A, B_2, W) → poly5, w_5, a_3, b_1
  97. Rotate(T, 15°, C) → T'
  98. Angle(T, C, T') → ε
  99. Ray(C, T') → q_1
  100. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  101. Vector(C, C'_8) → u_2
  102. Dilate(D, If[23 < e ≤ 24, 24 - e, 0], W) → D'_3
  103. Dilate(C, If[23 < e ≤ 24, 24 - e, 0], W) → C'_9
  104. Dilate(B_2, If[25 < e ≤ 26, 26 - e, 0], W) → G_1
  105. Dilate(A, If[25 < e ≤ 26, 26 - e, 0], W) → A'_2
  106. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  107. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  108. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  109. Midpoint(A, D) → C_1
  110. Midpoint(D, R) → D_1
  111. Rotate(A, ((-π)) / 2, B_2) → A'_3
  112. Rotate(B_2, π / 2, A) → E_1
  113. Midpoint(A'_3, E_1) → F_1
  114. Segment(F_1, A) → n_1
  115. Segment(C_1, B_2) → p_1
  116. Segment(B_2, F_1) → r_1
  117. Intersect(n_1, p_1) → J_1
  118. Segment(F_1, J_1) → s_1
  119. Circle(F_1, s_1 / 3) → c_1
  120. Intersect(c_1, n_1, 1) → K_1
  121. Circle(K_1, s_1 / 3) → e_1
  122. Intersect(e_1, n_1, 2) → L_1
  123. Circle(J_1, s_1 / 3) → t_1
  124. Intersect(t_1, p_1, 2) → M_1
  125. Circle(M_1, s_1 / 3) → e_2
  126. Intersect(e_2, p_1, 2) → N_1
  127. Circle(N_1, s_1 / 3) → f_2
  128. Intersect(f_2, p_1, 2) → O_1
  129. Intersect(c_1, r_1) → P_1,
  130. Circle(P_1, s_1 / 3) → k_2
  131. Intersect(k_2, r_1, 1) → Q_1
  132. Circle(Q_1, s_1 / 3) → p_2
  133. Intersect(p_2, r_1, 1) → R_1
  134. Circle(R_1, s_1 / 3) → q_2
  135. Intersect(q_2, r_1, 1) → S_1
  136. Dilate(B_2, If[14 < e ≤ 15, 15 - e, 0], O_1) → T_1
  137. Dilate(B_2, If[14 < e ≤ 15, 15 - e, 0], S_1) → U_1
  138. Segment(B_2, T_1) → i_2
  139. Segment(B_2, U_1) → m_2
  140. Dilate(O_1, If[15 < e ≤ 16, 16 - e, 0], N_1) → V_1
  141. Dilate(S_1, If[15 < e ≤ 16, 16 - e, 0], R_1) → W_1
  142. Segment(O_1, V_1) → n_2
  143. Segment(S_1, W_1) → r_2
  144. Dilate(A, If[15 < e ≤ 16, 16 - e, 0], J_1) → A'_4
  145. Segment(A, A'_4) → s_2
  146. Dilate(A, If[10 < e ≤ 11, 11 - e, 0], C_1) → A'_5
  147. Segment(A, A'_5) → t_2
  148. Rotate(A, If[11 < e ≤ 12, ((e - 11) * π), π], C_1) → A'_6
  149. Rotate(A'_5, If[11 < e ≤ 12, ((e - 11) * π / 2), π / 2], C_1) → Z_1
  150. Segment(A'_6, Z_1) → f_3
  151. Rotate(A'_6, If[12 < e ≤ 13, ((e - 12) * π / 2), π / 2], D) → A_2
  152. Rotate(Z_1, If[12 < e ≤ 13, ((e - 12) * π / 2), π / 2], D) → C_2
  153. Segment(A_2, C_2) → g_3
  154. Rotate(A_2, If[13 < e ≤ 14, ((e - 13) * π), π], F_1) → D_2
  155. Rotate(C_2, If[13 < e ≤ 14, ((e - 13) * π), π], F_1) → E_2
  156. Segment(D_2, E_2) → j_3
  157. Polygon(A, F_1, D) → t1, d_4, a_4, f_4
  158. Polygon(F_1, B_2, A'_1) → t2, a'_1, f_5, b_3
  159. Polygon(A, J_1, B_2) → t3, b_4, a_5, j_4
  160. Polygon(F_1, J_1, B_2) → t4, b_5, f_6, j_5
  161. Dilate(N_1, If[16 < e ≤ 17, 17 - e, 0], M_1) → F_2
  162. Dilate(R_1, If[16 < e ≤ 17, 17 - e, 0], Q_1) → G_2
  163. Segment(N_1, F_2) → k_3
  164. Segment(R_1, G_2) → l_3
  165. Dilate(M_1, If[17 < e ≤ 18, 18 - e, 0], J_1) → H_2
  166. Dilate(Q_1, If[17 < e ≤ 18, 18 - e, 0], P_1) → I_2
  167. Segment(M_1, H_2) → m_3
  168. Segment(Q_1, I_2) → n_3
  169. Dilate(J_1, If[16 < e ≤ 17, 17 - e, 0], L_1) → J_2
  170. Segment(J_1, J_2) → p_3
  171. Dilate(L_1, If[17 < e ≤ 18, 18 - e, 0], K_1) → K_2
  172. Segment(L_1, K_2) → q_3
  173. Dilate(J_1, If[18 < e ≤ 19, 19 - e, 0], C_1) → L_2
  174. Dilate(K_1, If[18 < e ≤ 19, 19 - e, 0], F_1) → M_2
  175. Dilate(P_1, If[18 < e ≤ 19, 19 - e, 0], F_1) → N_2
  176. Segment(J_1, L_2) → r_3
  177. Segment(K_1, M_2) → s_3
  178. Segment(P_1, N_2) → t_3
  179. Rotate(K_1, If[20 < e ≤ 21, ((-(e - 20)) * π), (-π)], K_1) → O_2
  180. Rotate(M_2, If[20 < e ≤ 21, ((-(e - 20)) * π), (-π)], K_1) → P_2
  181. Segment(O_2, P_2) → e_3
  182. Rotate(O_2, If[21 < e ≤ 22, ((-(e - 21)) * π), (-π)], L_1) → Q_2
  183. Rotate(P_2, If[21 < e ≤ 22, ((-(e - 21)) * π), (-π)], L_1) → R_2
  184. Segment(Q_2, R_2) → g_4
  185. Rotate(Q_2, If[22 < e ≤ 23, ((-(e - 22)) * π / 2), ((-π)) / 2], J_1) → S_2
  186. Rotate(R_2, If[22 < e ≤ 23, ((-(e - 22)) * π / 2), ((-π)) / 2], J_1) → T_2
  187. Segment(S_2, T_2) → h_4
  188. Rotate(S_2, If[23 < e ≤ 24, ((-(e - 23)) * π), (-π)], M_1) → U_2
  189. Rotate(T_2, If[23 < e ≤ 24, ((-(e - 23)) * π), (-π)], M_1) → V_2
  190. Segment(U_2, V_2) → i_4
  191. Rotate(U_2, If[24 < e ≤ 25, ((-(e - 24)) * π), (-π)], N_1) → W_2
  192. Rotate(V_2, If[24 < e ≤ 25, ((-(e - 24)) * π), (-π)], N_1) → Z_2
  193. Segment(W_2, Z_2) → k_4
  194. Rotate(W_2, If[25 < e ≤ 26, ((-(e - 25)) * π), (-π)], O_1) → A_3
  195. Rotate(Z_2, If[25 < e ≤ 26, ((-(e - 25)) * π), (-π)], O_1) → B_3
  196. Segment(A_3, B_3) → j_2
  197. Rotate(A_3, If[26 < e ≤ 27, ((-(e - 26)) * atand(3 / 4)), (-atand(3 / 4))], B_2) → C_3
  198. Rotate(B_3, If[26 < e ≤ 27, ((-(e - 26)) * atand(3 / 4)), (-atand(3 / 4))], B_2) → D_3
  199. Segment(C_3, D_3) → l_2
  200. Rotate(C_3, If[27 < e ≤ 28, ((-(e - 27)) * π), (-π)], S_1) → E_3
  201. Rotate(D_3, If[27 < e ≤ 28, ((-(e - 27)) * π), (-π)], S_1) → F_3
  202. Segment(E_3, F_3) → h_3
  203. Rotate(E_3, If[28 < e ≤ 29, ((-(e - 28)) * π), (-π)], R_1) → G_3
  204. Rotate(F_3, If[28 < e ≤ 29, ((-(e - 28)) * π), (-π)], R_1) → H_3
  205. Segment(G_3, H_3) → i_3
  206. Rotate(G_3, If[29 < e ≤ 30, ((-(e - 29)) * π), (-π)], Q_1) → I_3
  207. Rotate(H_3, If[29 < e ≤ 30, ((-(e - 29)) * π), (-π)], Q_1) → J_3
  208. Segment(I_3, J_3) → l_4
  209. Rotate(I_3, If[30 < e ≤ 31, ((-(e - 30)) * π), (-π)], P_1) → K_3
  210. Rotate(J_3, If[30 < e ≤ 31, ((-(e - 30)) * π), (-π)], P_1) → L_3
  211. Segment(K_3, L_3) → m_4
  212. Polygon(J_1, B_2, F_1) → t5, f_7, j_6, b_6