Proof of Triangle Numbers Without Words

GeoGebra Commands & Parameters

1. Polygon(A, B, 3) → poly1, f, g, h, C
2. Point(poly1) → D
3. Point(poly1) → E
4. Polygon(A, D, 3) → poly2, i, j, k, F
5. Polygon(A, E, 3) → poly3, l, m, n, G
6. Segment(H, I) → p
7. Segment(H_1, I_1) → p_1
8. Polygon(H, I, 4) → poly4, q, r, s, t_3, J, K
9. Polygon(H_1, I_1, 4) → poly5, a, b, c, d, L, M
10. Segment(H_2, I_2) → p_2
11. Polygon(H_2, I_2, 4) → poly4_1, q_1, r_1, s_1, t_1, J_1, K_1
12. Segment(H_3, I_3) → p_3
13. Polygon(H_3, I_3, 4) → poly5_1, a_1, b_1, c_1, d_1, L_1, M_1
14. Segment(H, H_2) → e
15. Segment(K, K_1) → f_1
16. Segment(I, I_2) → g_1
17. Segment(J, J_1) → h_1
18. Segment(H_1, H_3) → i_1
19. Segment(M, M_1) → j_1
20. Segment(I_1, I_3) → k_1
21. Segment(L, L_1) → l_1
22. Point(f) → N
23. Line(N, j) → m_1
24. Point(f) → O
25. Line(O, m) → n_1
26. Point(n_1) → P
27. Point(n_1) → Q
28. Point(m_1) → R
29. Point(m_1) → S
30. Line(T, n_1) → e_1
31. Line(U, m_1) → f_2
32. Line(S, f) → g_2
33. Line(Q, f) → h_2
34. Intersect(g_2, f_2) → V
35. Intersect(h_2, e_1) → W
36. Intersect(g_2, poly1) → Z, A_1
37. Intersect(g_2, e_1) → B_1
38. Polygon(R, U, V, S) → q1, r_2, u, v, s_2
39. Polygon(P, T, W, Q) → q2, p_4, t_2, w, q_2
40. Segment(R, S) → i_2
41. Segment(P, Q) → j_2
42. Intersect(h, q_2) → C_1
43. Intersect(h, t_2) → D_1
44. Polygon(C_1, D_1, Q) → t1, q_3, c_2, d_2
45. Polygon(A_2, B_2, 3) → poly1_1, f_3, g_3, h_3, C_3
46. Point(poly1_1) → D_3
47. Point(poly1_1) → E_1
48. Polygon(A_2, E_1, 3) → poly3_1, l_2, m_2, n_3, G_1
49. Point(f_3) → N_1
50. Point(f_3) → O_1
51. Line(O_1, m_2) → n_2
52. Point(n_2) → P_1
53. Line(T_1, n_2) → e_2
54. Polygon(A_3, B_3, 3) → poly1_2, f_4, g_4, h_5, C_5
55. Point(poly1_2) → D_5
56. Point(poly1_2) → E_2
57. Polygon(A_3, E_2, 3) → poly3_2, l_3, m_3, n_5, G_2
58. Point(f_4) → N_2
59. Point(f_4) → O_2
60. Line(O_2, m_3) → n_4
61. Point(n_4) → P_2
62. Line(T_2, n_4) → e_3
63. Polygon(C_1, D_1, Q) → t2, q_4, c_3, d_3
64. Rotate(C_1, (3 * (t - 3)), S_1) → F_1
65. Rotate(D_1, (3 * (t - 3)), S_1) → Q_1
66. Rotate(Q, (3 * (t - 3)), S_1) → Q'
67. Polygon(F_1, Q_1, Q') → t2', q', f_5, q_5
68. Polygon(I_2, I, J, J_1) → q3, i_3, i_4, j_3, j_4
69. Polygon(I_1, L, L_1, I_3) → q4, i_5, l_4, l_5, i_6
70. Polygon(H_1, I_1, L, L_1, M_1, H_3) → poly6, h_4, i_7, l_6, l_7, m_4, h_6
71. Polygon(H, I, J, J_1, K_1, H_2) → poly7, h_7, i_8, j_5, j_6, k_2, h_8
72. Vector(R_1, U_1) → u_1
73. Point(u_1) → V_1
74. Vector(V_1, W_1) → w_1
75. Vector(R_2, U_2) → u_2
76. Vector(Z_1, C_2) → v_1
77. Point(v_1) → D_2
78. Vector(D_2, F_2) → v_2
79. Vector(Z_2, C_4) → v_4
80. Vector(J_2, K_2) → w_2
81. Point(w_2) → L_2
82. Vector(L_2, M_2) → a_2
83. Vector(R_3, U_3) → u_3
84. Point(u_3) → V_2
85. Vector(V_2, W_2) → w_3
86. Vector(R_4, U_4) → u_4
87. Point(u_4) → V_3