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无字证明：x²+ax= (x+a/2)²-(a/2)²

无字证明：x²+ax= (x+a/2)²-(a/2)²
几何画板及绘图内容加载中，请稍等……
2020-04-17 04:26
函数解析几何几何作图
①本范例通过动态图形演示了代数恒等式x²+ax= (x+a/2)²-(a/2)²的几何意义；②涉及核心数学概念包括完全平方公式、代数变形、几何直观；③作图要点运用了Circle（圆）、Point（点）、Polygon（多边形）、Line（线）、OrthogonalLine（垂线）、Intersect（交点）、Midpoint（中点）、Vector（向量）、Translate（平移）、Rotate（旋转）、Segment（线段）等GeoGebra命令；④适合高中数学学生用于理解代数与几何的联系，提升可视化数学能力。

GeoGebra 命令及参数

  1. Circle(A, c) → c_1
  2. Point(c_1) → B
  3. Polygon(A, B, 4) → poly1, a_1, b, d, e, C, D
  4. Line(E, a_1) → g
  5. OrthogonalLine(E, g) → i
  6. Circle(E, a) → h
  7. Intersect(h, i) → H, I
  8. Midpoint(I, E) → L
  9. Vector(L, N) → v
  10. Translate(L, Vector[(If[Animation < 5, Animation / 5, 1] * v)]) → L'
  11. Vector(A, D_2) → w_1
  12. Circle(E, c) → f
  13. Intersect(f, g) → F, G
  14. OrthogonalLine(G, g) → j
  15. Line(I, g) → k
  16. Intersect(k, j) → J
  17. Polygon(E, G, J, I) → poly2, e_1, g_1, j_1, i_1
  18. Translate(A, Vector[(If[Animation < 5, Animation / 5, 1] * w_1)]) → K
  19. Vector(A, K) → u
  20. Translate(poly1, u) → poly1'
  21. Midpoint(J, G) → M
  22. Polygon(L, M, J, I) → poly3, l, m, j_2, i_2
  23. Polygon(L, M, G, E) → poly4, l_1, m_1, g_2, e_2
  24. Vector(E, O) → w
  25. Translate(M, Vector[(If[Animation < 5, Animation / 5, 1] * v)]) → M'
  26. Translate(J, Vector[(If[Animation < 5, Animation / 5, 1] * v)]) → J'
  27. Translate(I, Vector[(If[Animation < 5, Animation / 5, 1] * v)]) → I'
  28. Polygon(L', M', J', I') → poly3', l', m', j', i'
  29. Translate(L, Vector[(If[Animation < 5, Animation / 5, 1] * w)]) → L'_1
  30. Translate(M, Vector[(If[Animation < 5, Animation / 5, 1] * w)]) → M'_1
  31. Translate(G, Vector[(If[Animation < 5, Animation / 5, 1] * w)]) → G'
  32. Translate(E, Vector[(If[Animation < 5, Animation / 5, 1] * w)]) → E'
  33. Polygon(L'_1, M'_1, G', E') → poly4', l'_1, m'_1, g', e'
  34. Vector(K, P) → z
  35. Translate(poly1', n) → poly1''
  36. Translate(poly1'', z) → poly1'''
  37. Vector(I', P) → p
  38. Translate(L', Vector[(If[Animation < 10, (Animation - 5) / 5, 1] * p)]) → L''
  39. Translate(M', Vector[(If[Animation < 10, (Animation - 5) / 5, 1] * p)]) → M''
  40. Translate(J', Vector[(If[Animation < 10, (Animation - 5) / 5, 1] * p)]) → J''
  41. Translate(I', Vector[(If[Animation < 10, (Animation - 5) / 5, 1] * p)]) → I''
  42. Polygon(L'', M'', J'', I'') → poly3'', l'', m'', j'', i''
  43. Vector(L'_1, J'') → q
  44. Rotate(L'_1, Animation, L'_1) → Q
  45. Rotate(L'_1, If[Animation < 7.84, Animation, 7.84], L'_1) → R
  46. Rotate(M'_1, If[Animation < 7.85, Animation, 7.85], L'_1) → S
  47. Rotate(G', If[Animation < 7.85, Animation, 7.85], L'_1) → G''
  48. Rotate(E', If[Animation < 7.85, Animation, 7.85], L'_1) → E''
  49. Polygon(R, S, G'', E'') → poly4'', r, s, g'', e''
  50. Translate(R, Vector[(If[Animation < 10, (Animation - 5) / 5, 1] * q)]) → R'
  51. Translate(S, Vector[(If[Animation < 10, (Animation - 5) / 5, 1] * q)]) → S'
  52. Translate(G'', Vector[(If[Animation < 10, (Animation - 5) / 5, 1] * q)]) → G'''
  53. Translate(E'', Vector[(If[Animation < 10, (Animation - 5) / 5, 1] * q)]) → E'''
  54. Polygon(R', S', G''', E''') → poly4''', r', s', g''', e'''
  55. Translate(L'_1, Vector[(If[Animation < 10, (Animation - 5) / 5, 1] * q)]) → T
  56. Translate(M'_1, Vector[(If[Animation < 10, (Animation - 5) / 5, 1] * q)]) → U
  57. Translate(G', Vector[(If[Animation < 10, (Animation - 5) / 5, 1] * q)]) → G''_1
  58. Translate(E', Vector[(If[Animation < 10, (Animation - 5) / 5, 1] * q)]) → E''_1
  59. Polygon(T, U, G''_1, E''_1) → poly4''_1, t, u_1, g''_1, e''_1
  60. Vector(J'', V) → v_1
  61. Polygon(J'', M'', 4) → poly5, b_1, d_1, f_1, h_1, W, Z
  62. Translate(poly5, Vector[(If[Animation < 15, (Animation - 10) / 10, 1] * v_1)]) → poly5'
  63. Translate(poly5, Vector[(If[Animation < 15, (Animation - 10) / 10, 1] * v_1)]) → poly5'_1
  64. Segment(D, C) → k_1
  65. Point(a_1) → B_1
  66. Segment(D, B_1) → p_1
  67. Midpoint(k_1) → C_1
  68. Midpoint(p_1) → D_1
  69. Translate(D, u) → D'
  70. Translate(C, u) → C'
  71. Segment(D', C') → q_1
  72. Translate(C, u) → C'_1
  73. Translate(A, u) → A'
  74. Translate(B, u) → B'
  75. Segment(A', B') → s_1
  76. Translate(D, u) → D'_1
  77. Translate(B_1, u) → F_1
  78. Segment(D'_1, F_1) → t_1
  79. Translate(D', z) → D''
  80. Translate(C', z) → C''
  81. Segment(D'', C'') → a_2
  82. Translate(D'_1, z) → G_1
  83. Translate(F_1, z) → H_1
  84. Segment(G_1, H_1) → b_2
  85. Translate(C'_1, z) → I_1
  86. Translate(A', z) → A''
  87. Translate(B', z) → B''
  88. Segment(A'', B'') → d_2
  89. Midpoint(q_1) → K_1
  90. Midpoint(a_2) → M_1
  91. Midpoint(b_2) → N_1
  92. Midpoint(t_1) → L_1
  93. Midpoint(e_1) → O_1
  94. Midpoint(i') → P_1
  95. Midpoint(e') → Q_1
  96. Midpoint(i'') → R_1
  97. Midpoint(l'') → S_1
  98. Midpoint(g''') → T_1
  99. Midpoint(s') → U_1
  100. Midpoint(l') → V_1
  101. Midpoint(g') → W_1
  102. Segment(C, B) → n_1
  103. Translate(C, u) → C'_2
  104. Translate(B, u) → B'_1
  105. Segment(C'_2, B'_1) → r_1
  106. Midpoint(C, B_1) → A_1
  107. Midpoint(D'', J'') → E_1
  108. Centroid(poly5') → J_1
  109. Centroid(poly2) → Z_1
  110. Centroid(poly1'') → A_2
  111. Centroid(poly3'') → B_2
  112. Centroid(poly4''') → C_2
  113. Segment(C'', B'') → f_2

AI 绘图提示语参考

画一个动态几何图形，展示代数恒等式 x²+ax= (x+a/2)²-(a/2)²。首先绘制一个正方形和一条对角线，然后以对角线中点为圆心画圆，交点连接形成新的图形。接着，通过旋转和平移操作，将图形变换为展示恒等式的两部分，包括旋转的正方形和移动的三角形，最终形成对称的几何证明。
以上提示语为根据原始绘图命令整理而来，仅供参考。