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Consecutive Triangle Numbers Form a Square

Consecutive Triangle Numbers Form a Square
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2015-09-24 16:59
TrianglePolygonSequenceGeometric ConstructionSlider Animation
① This GeoGebra example dynamically demonstrates that the sum of two consecutive triangular numbers always equals a square number, by allowing users to move triangular dot patterns. ② The core mathematical concept is the relationship between triangular numbers (T_n = n(n+1)/2) and square numbers, specifically T_n + T_{n+1} = (n+1)^2, illustrating a beautiful number-theoretic identity through visual geometry. ③ Construction details: The applet uses GeoGebra commands such as Point, Segment, and Polygon to create multiple triangular arrays (e.g., poly1, poly1_1, poly2, poly3) with many points and segments. Users can drag points (like B, C, D, E) or use sliders to move the triangles, observing how two consecutive triangular numbers fit together to form a square. ④ This resource is suitable for middle or high school students learning about sequences, figurate numbers, or algebraic identities, and is ideal for exploratory activities in mathematics classrooms to foster visual and conceptual understanding.

GeoGebra Commands & Parameters

  1. Point(Circle[B, 1]) → C
  2. Segment(B, C) → a
  3. Polygon(B, C, 4) → poly1, b, c, d, e, D, E
  4. Point(Circle[B_1, 1]) → C_1
  5. Segment(B_1, C_1) → a_1
  6. Polygon(B_1, C_1, 4) → poly1_1, b_1, c_1, d_1, e_1, D_1, E_1
  7. Polygon(E_1, D_1, 4) → poly2, f, g, h, i, F, G
  8. Polygon(F, D_1, 4) → poly3, j, k, l, m, H, I
  9. Point(Circle[B_2, 1]) → C_2
  10. Segment(B_2, C_2) → a_2
  11. Polygon(B_2, C_2, 4) → poly1_2, b_2, c_2, d_2, e_2, D_2, E_2
  12. Polygon(E_2, D_2, 4) → poly2_1, f_1, g_1, h_1, i_1, F_1, G_1
  13. Polygon(F_1, D_2, 4) → poly3_1, j_1, k_1, l_1, m_1, H_1, I_1
  14. Polygon(E_2, G_1, 4) → poly4, n, p, q, r, J, K
  15. Polygon(B_2, E_2, 4) → poly5, s, t, n_1, p_1, L, M
  16. Polygon(B_2, M, 4) → poly6, q_1, r_1, s_1, t_1, N, O
  17. Point(Circle[B_3, 1]) → C_3
  18. Segment(B_3, C_3) → a_3
  19. Polygon(B_3, C_3, 4) → poly1_3, b_3, c_3, d_3, e_3, D_3, E_3
  20. Polygon(E_3, D_3, 4) → poly2_2, f_2, g_2, h_2, i_2, F_2, G_2
  21. Polygon(F_2, D_3, 4) → poly3_2, j_2, k_2, l_2, m_2, H_2, I_2
  22. Polygon(E_3, G_2, 4) → poly4_1, n_2, p_2, q_2, r_2, J_1, K_1
  23. Polygon(B_3, E_3, 4) → poly5_1, s_2, t_2, n_3, p_3, L_1, M_1
  24. Polygon(B_3, M_1, 4) → poly6_1, q_3, r_3, s_3, t_3, N_1, O_1
  25. Polygon(N_1, M_1, 4) → poly7, f_3, g_3, h_3, i_3, P, Q
  26. Polygon(M_1, K_1, 4) → poly8, j_3, k_3, l_3, m_3, R, S
  27. Polygon(K_1, J_1, 4) → poly9, a_4, b_4, c_4, d_4, T, U
  28. Polygon(N_1, Q, 4) → poly10, e_4, f_4, g_4, h_4, V, W
  29. Point(Circle[B_4, 1]) → C_4
  30. Segment(B_4, C_4) → a_5
  31. Polygon(B_4, C_4, 4) → poly1_4, b_5, c_5, d_5, e_5, D_4, E_4
  32. Polygon(E_4, D_4, 4) → poly2_3, f_5, g_5, h_5, i_4, F_3, G_3
  33. Polygon(F_3, D_4, 4) → poly3_3, j_4, k_4, l_4, m_4, H_3, I_3
  34. Polygon(E_4, G_3, 4) → poly4_2, n_4, p_4, q_4, r_4, J_2, K_2
  35. Polygon(B_4, E_4, 4) → poly5_2, s_4, t_4, n_5, p_5, L_2, M_2
  36. Polygon(B_4, M_2, 4) → poly6_2, q_5, r_5, s_5, t_5, N_2, O_2
  37. Polygon(N_2, M_2, 4) → poly7_1, f_6, g_6, h_6, i_5, P_1, Q_1
  38. Polygon(M_2, K_2, 4) → poly8_1, j_5, k_5, l_5, m_5, R_1, S_1
  39. Polygon(K_2, J_2, 4) → poly9_1, a_6, b_6, c_6, d_6, T_1, U_1
  40. Polygon(N_2, Q_1, 4) → poly10_1, e_6, f_7, g_7, h_7, V_1, W_1
  41. Polygon(V_1, Q_1, 4) → poly11, i_6, j_6, k_6, l_6, Z, A_1
  42. Polygon(Q_1, P_1, 4) → poly12, m_6, n_6, p_6, q_6, Z_1, A_2
  43. Polygon(P_1, R_1, 4) → poly13, r_6, s_6, t_6, a_7, P_2, Q_2
  44. Polygon(R_1, T_1, 4) → poly14, b_7, c_7, d_7, e_7, R_2, S_2
  45. Polygon(V_1, A_1, 4) → poly15, i_7, j_7, k_7, l_7, T_2, U_2

AI Drawing Prompt Reference

Create a dynamic geometric figure composed of multiple square grids, demonstrating that the sum of two consecutive triangular numbers equals a square number. The figure consists of several nested layers of squares, each layer containing multiple small squares, arranged in a stepped pattern. Users can drag control points or use sliders to adjust the positions of layers, observing how two triangular arrays fit together to form a complete large square. Use distinct colors for different square regions and maintain neat dot arrangements to visually illustrate the number-theoretic identity.
The prompt above was compiled from the original drawing commands and is for reference only.