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Proof of a^3 - b^3 without words

Proof of a^3 - b^3 without words
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2019-12-28 07:07
Coordinate GeometrySolid GeometryGeometric Construction
In collections Algebraic puzzles for binomial formula
This example visually demonstrates the proof of the cube difference formula a^3 - b^3 using geometric shapes. Key mathematical concepts include cube difference, geometric construction, and coordinate geometry. Using GeoGebra commands such as PointIn, Cube, Polygon, and Vector, a 3D model is constructed to illustrate the geometric meaning of the cube difference formula. Suitable for students at the junior high school level and above.

GeoGebra Commands & Parameters

  1. PointIn(f) → A
  2. Point(Circle[B, Distance[C, B], Segment[C, B]]) → L
  3. Cube(C, B, L) → c, M, faceLMQP, faceNOPQ, arêteCM_1, arêteBC, arêteBL, arêteLM, arêteCN, arêteMQ, arêteNQ, arêteNO, N, arêteBO, arêteOP_1, arêteLP, arêtePQ, O, P, Q, faceBCML, faceCMQN, faceBCNO, faceBLPO
  4. Polygon(E, L, M, C, A, D) → poly1, e, l, m, c_1, a_1, d
  5. Polygon(T, U, O, N, C, A) → poly2, t, u, o_3, n, c_2, a_2
  6. Polygon(U, R, E, L, P, O) → poly3, u_1, r, e_1, l_1, p, o_1
  7. Polygon(P, O, N, Q) → q1, p_1, o_2, n_1, q
  8. Polygon(Q, N, C, M) → q2, q_1, n_2, c_3, m_1
  9. Polygon(R, S, D, E) → q3, r_1, s, d_1, e_2
  10. Polygon(S, T, A, D) → q4, s_1, t_1, a_3, d_2
  11. Polygon(U, T, S, R) → q5, u_2, t_2, s_2, r_2
  12. PointIn(f) → V
  13. Vector(C, V) → v
  14. Translate(c, Vector[(If[Animation ≤ 10, Animation / 10, 1] * v)]) → c'
  15. Polygon(Q, P, L, M) → q6, q_2, p_2, l_2, m_2
  16. Segment(W, C_1) → g
  17. Segment(C_1, A_1) → h
  18. Segment(B_1, Z) → i
  19. Segment(Z, O) → j
  20. Segment(W, L) → k
  21. Prism(L, A_1, D, E, W) → w, D_1, arête4, arête1, arêteDE, arêteEL, arêteLW, arête7, arête2, arête3, arête5, arête8, E_1, arête9, arête6, F_1, face1, face5, face3, face2, face4, face6
  22. Prism(B_1, Z, O, P, D_1) → b_1, G_1, arête13, arêteOZ, arêteOP, arête11, arête15, arête14, arête10, arête12, arête16, arête18, H_1, arête19, arête17, I_1, face7, face11, face9, face8, face10, face12
  23. Prism(A_1, A, C, M, B_1) → f_1, J_1, arête20, arêteAC, arêteCM, arête23, arête25, arête21, arête22, arête24, arête26, arête28, K_1, arête29, arête27, L_1, face13, face15, face14, face16, face17, face18
  24. PointIn(f) → M_1
  25. Vector(D, M_1) → v_1
  26. PointIn(f) → N_1
  27. PointIn(f) → O_1
  28. Vector(B_1, N_1) → w_1
  29. Translate(b_1, Vector[((Animation - 10) / 10 * w_1)]) → g_1
  30. Vector(A_1, O_1) → h_1
  31. Translate(f_1, Vector[((Animation - 10) / 10 * h_1)]) → i_1
  32. Translate(c, Vector[(If[Animation ≤ 10, Animation / 10, 1] * v)]) → c'_1
  33. Translate(b_1, Vector[((Animation - 10) / 10 * w_1)]) → j_1
  34. Translate(w, Vector[((Animation - 10) / 10 * v_1)]) → w'_1
  35. Translate(f_1, Vector[((Animation - 10) / 10 * h_1)]) → k_1
  36. Translate(w, Vector[((Animation - 10) / 10 * v_1)]) → w'
  37. Midpoint(R_1, S_1) → Z_1
  38. Vector(Z_1, S_1) → v_2
  39. Vector(Z_1, R_1) → w_2
  40. Midpoint(T_1, U_1) → A_2
  41. Vector(A_2, U_1) → b_2
  42. Vector(A_2, T_1) → f_2
  43. Midpoint(B_2, Q_1) → C_2
  44. Vector(C_2, Q_1) → g_2
  45. Vector(C_2, B_2) → h_2
  46. Midpoint(D_2, B_2) → E_2
  47. Vector(E_2, B_2) → i_2
  48. Vector(E_2, D_2) → j_2
  49. Midpoint(P_1, Q_1) → F_2
  50. Vector(F_2, P_1) → k_2
  51. Vector(F_2, Q_1) → u_3
  52. Midpoint(W_1, V_1) → G_2
  53. Vector(G_2, W_1) → v_3
  54. Vector(G_2, V_1) → w_3
  55. Segment(C, B) → f_3
  56. Segment(L, B) → g_3
  57. Segment(O, B) → h_3
  58. Midpoint(M_2, N_2) → W_2
  59. Vector(W_2, N_2) → b_3
  60. Vector(W_2, M_2) → d_3
  61. Midpoint(P_2, Q_2) → Z_2
  62. Midpoint(U_2, V_2) → A_3
  63. Midpoint(K_2, J_2) → B_3
  64. Vector(B_3, K_2) → e_3
  65. Vector(B_3, J_2) → i_3
  66. Vector(Z_2, Q_2) → j_3
  67. Vector(Z_2, P_2) → k_3
  68. Midpoint(R_2, S_2) → C_3
  69. Midpoint(T_2, R_2) → D_3
  70. Midpoint(H_2, I_2) → E_3
  71. Vector(E_3, I_2) → l_3
  72. Vector(E_3, H_2) → m_3
  73. Vector(D_3, T_2) → n_3
  74. Vector(D_3, R_2) → p_3
  75. Vector(C_3, S_2) → q_3
  76. Vector(C_3, R_2) → r_3
  77. Vector(A_3, V_2) → s_3
  78. Vector(A_3, U_2) → t_3
  79. Midpoint(J_2, L_2) → F_3
  80. Vector(F_3, L_2) → u_4
  81. Vector(F_3, J_2) → v_4
  82. Midpoint(O_2, H_2) → G_3
  83. Vector(G_3, O_2) → w_4
  84. Vector(G_3, H_2) → a_4
  85. Midpoint(U_1, H_3) → I_3
  86. Vector(I_3, H_3) → b_4
  87. Vector(I_3, U_1) → c_4
  88. PointIn(w') → J_3
  89. PointIn(k_1) → K_3
  90. PointIn(j_1) → L_3

AI Drawing Prompt Reference

Create a geometric proof of the cube difference formula a^3 - b^3, showing the difference between two cubes. Start by constructing a cube, then use geometric shapes and coordinate geometry to break down the cube difference into multiple parts, represented by polygons and lines in different colors. The final image should include various geometric bodies and vectors, clearly demonstrating the geometric meaning of the cube difference formula, suitable for junior high school students and above.
The prompt above was compiled from the original drawing commands and is for reference only.