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Wordless Proof: x²+ax= (x+a/2)²-(a/2)²

Wordless Proof: x²+ax= (x+a/2)²-(a/2)²
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2020-04-17 04:26
FunctionCoordinate GeometryGeometric Construction
This example visually demonstrates the algebraic identity x²+ax= (x+a/2)²-(a/2)² through dynamic geometry. Key math concepts include perfect square formula, algebraic manipulation, and geometric intuition. GeoGebra commands used in construction are Circle, Point, Polygon, Line, OrthogonalLine, Intersect, Midpoint, Vector, Translate, Rotate, and Segment. Ideal for high school students to understand the connection between algebra and geometry, enhancing visual mathematical reasoning.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Create a dynamic geometry visualization demonstrating the algebraic identity x²+ax= (x+a/2)²-(a/2)². Start by drawing a square and its diagonal, then draw a circle centered at the midpoint of the diagonal. Connect the intersection points to form a new shape. Next, rotate and translate parts of the figure to show the two sides of the identity, including a rotating square and a moving triangle, resulting in a symmetric geometric proof.
The prompt above was compiled from the original drawing commands and is for reference only.