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Edgardo Alandete's Proof of the Pythagorean Theorem #4

Edgardo Alandete's Proof of the Pythagorean Theorem #4
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2012-11-18 07:19
TriangleAngleCoordinate GeometryGeometric Construction
This example presents Edgardo Alandete's proof of the Pythagorean theorem, illustrating the relationship between the legs and hypotenuse of a right triangle through dynamic geometry. Key concepts include the Pythagorean theorem, triangle properties, and coordinate geometry. The construction uses GeoGebra commands such as Segment, Point, Polygon, and Intersect to create points and lines, with intersections verifying a² + b² = c². It is suitable for high‑school geometry classes and math contest preparation.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Segment(A, C) → b
  3. Segment(C, D) → c
  4. Segment(D, B) → d
  5. Segment(A, E) → e
  6. Segment(E, B) → f
  7. Segment(E, F) → g
  8. Segment(A, G) → h
  9. Segment(G, F) → i
  10. Segment(E, H) → j
  11. Segment(H, I) → k
  12. Segment(I, J) → l
  13. Segment(J, K) → m
  14. Segment(K, B) → n
  15. Segment(L, M) → p
  16. Segment(A, L) → q
  17. Segment(M, C) → r
  18. Segment(N, O) → s
  19. Segment(N, B) → t
  20. Segment(O, D) → a_1
  21. Point(f) → P
  22. Segment(P, Q) → b_1
  23. Segment(Q, A) → c_1
  24. Segment(N, R) → d_1
  25. Segment(M, S) → e_1
  26. Point(e_1) → T
  27. Segment(Q, T) → f_1
  28. Point(d_1) → U
  29. Segment(U, S) → g_1
  30. Segment(V, W) → h_1
  31. Segment(V, C) → i_1
  32. Segment(W, S) → j_1
  33. Segment(D, W) → k_1
  34. Point(j) → Z
  35. Point(m) → A_1
  36. Segment(Z, A_1) → l_1
  37. Point(l_1) → B_1
  38. Segment(P, B_1) → m_1
  39. Point(l) → C_1
  40. Segment(B_1, C_1) → n_1
  41. Point(l_1) → D_1
  42. Segment(I, D_1) → p_1
  43. Point(n_1) → E_1
  44. Segment(J, E_1) → q_1
  45. Point(m_1) → F_1
  46. Segment(E, F_1) → r_1
  47. Point(l_1) → G_1
  48. Segment(B, G_1) → s_1
  49. Polygon(A, L, T, Q) → poly1, a_2, l_2, t_1, q_2
  50. Polygon(C_1, I, Z, B_1) → poly2, c_2, i_2, z, b_2
  51. Intersect(b, p) → H_1
  52. Polygon(C, M, H_1) → poly3, h_2, c_3, m_2
  53. Polygon(N, R, P, B) → poly4, n_2, r_2, p_2, b_3
  54. Intersect(a, b_1) → I_1
  55. Polygon(I_1, A, Q) → poly5, q_3, i_3, a_3
  56. Polygon(A_1, B_1, C_1, J) → poly6, a_4, b_4, c_4, j_2
  57. Polygon(D, W, U, O) → poly7, d_2, w, u, o
  58. Intersect(d, s) → J_1
  59. Polygon(B, N, J_1) → poly8, j_3, b_5, n_3
  60. Polygon(B_1, P, B, A_1) → poly9, b_6, p_3, b_7, a_5
  61. Polygon(M, C, V, S) → poly10, m_3, c_5, v, s_2
  62. Intersect(c, h_1) → K_1
  63. Polygon(W, K_1, D) → poly11, d_3, w_1, k_2
  64. Polygon(Z, E, P, B_1) → poly12, z_1, e_2, p_4, b_8
  65. Polygon(E, F, G, A) → poly13, e_3, f_2, g_2, a_6
  66. Polygon(R, T, S, U) → poly14, r_3, t_2, s_3, u_1
  67. Polygon(A, H_1, L) → poly15, l_3, a_7, h_3
  68. Polygon(H_1, M, C) → poly16, c_6, h_4, m_4
  69. Polygon(C, V, K_1) → poly17, k_3, c_7, v_1
  70. Polygon(K_1, D, W) → poly18, w_2, k_4, d_4
  71. Polygon(E, P, F_1) → poly19, f_3, e_4, p_5
  72. Polygon(I, D_1, Z) → poly20, z_2, i_4, d_5
  73. Polygon(C_1, E_1, J) → poly21, j_4, c_8, e_5
  74. Polygon(P, B, I_1) → poly22, i_5, p_6, b_9
  75. Polygon(I_1, Q, A) → poly23, a_8, i_6, q_4
  76. Polygon(G_1, A_1, B) → poly24, b_{10}, g_3, a_9
  77. Polygon(B, N, J_1) → poly25, j_5, b_{11}, n_4
  78. Polygon(D, O, J_1) → poly26, j_6, d_6, o_1
  79. Segment(F, E) → t_3
  80. Segment(E, A) → f_4
  81. Segment(A, G) → g_4
  82. Segment(G, F) → l_4
  83. Segment(E, B) → r_4
  84. Segment(B, A) → s_4
  85. Segment(A, C) → t_4
  86. Segment(C, D) → f_5
  87. Segment(D, B) → g_5
  88. Segment(B, J) → h_5
  89. Segment(J, I) → k_5
  90. Segment(I, E) → l_5
  91. Segment(Z, A_1) → m_5
  92. Segment(P, C_1) → n_5
  93. Segment(S, L) → q_5
  94. Segment(T, P) → r_5
  95. Segment(R, O) → s_5
  96. Segment(U, V) → t_5
  97. Segment(L, A) → e_6
  98. Segment(C, V) → f_6
  99. Segment(D, O) → g_6
  100. Intersect(k, l) → L_1
  101. Intersect(l_1, p_1) → M_1
  102. Segment(L_1, M_1) → h_6
  103. Segment(M_1, L_1) → k_6
  104. Intersect(n_1, q_1) → N_1
  105. Intersect(l, m) → O_1
  106. Segment(N_1, O_1) → l_6
  107. Segment(O_1, N_1) → m_6
  108. Segment(N_1, O_1) → n_6
  109. Segment(O_1, N_1) → q_6
  110. Segment(M_1, L_1) → r_6
  111. Segment(L_1, M_1) → s_6
  112. Segment(N_1, O_1) → t_6
  113. Point(l_1) → P_1
  114. Intersect(a, d) → Q_1
  115. Segment(P_1, Q_1) → d_7
  116. Segment(Q_1, P_1) → e_7
  117. Segment(P_1, Q_1) → f_7
  118. Segment(Q_1, P_1) → g_7
  119. Segment(P_1, Q_1) → h_7
  120. Segment(Q_1, P_1) → i_7
  121. Intersect(m_1, r_1) → R_1
  122. Intersect(e, f) → S_1
  123. Segment(R_1, S_1) → j_7
  124. Segment(S_1, R_1) → k_7
  125. Segment(R_1, S_1) → l_7
  126. Segment(S_1, R_1) → m_7
  127. Segment(R_1, S_1) → n_7
  128. Segment(S_1, R_1) → p_7
  129. Segment(R_1, S_1) → q_7
  130. Segment(Q_1, P_1) → r_7
  131. Segment(P_1, Q_1) → s_7
  132. Segment(R_1, S_1) → t_7
  133. Segment(S_1, R_1) → d_8
  134. Segment(R_1, S_1) → e_8
  135. Segment(S_1, R_1) → f_8
  136. Intersect(d, t) → T_1
  137. Intersect(s, t) → U_1
  138. Segment(T_1, U_1) → g_8
  139. Segment(U_1, T_1) → h_8
  140. Segment(T_1, U_1) → i_8
  141. Segment(U_1, T_1) → j_8
  142. Intersect(b_1, c_1) → V_1
  143. Intersect(a, b) → W_1
  144. Segment(V_1, W_1) → k_8
  145. Segment(W_1, V_1) → l_8
  146. Segment(V_1, W_1) → m_8
  147. Segment(W_1, V_1) → n_8
  148. Intersect(c, d) → Z_1
  149. Intersect(h_1, j_1) → A_2
  150. Segment(Z_1, A_2) → p_8
  151. Segment(A_2, Z_1) → q_8
  152. Segment(Z_1, A_2) → r_8
  153. Intersect(a, d) → B_2
  154. Point(l_1) → C_2
  155. Segment(B_2, C_2) → s_8
  156. Point(l_1) → D_2
  157. Intersect(f, n) → E_2
  158. Segment(D_2, E_2) → t_8
  159. Intersect(l, m) → F_2
  160. Intersect(n_1, q_1) → G_2
  161. Segment(F_2, G_2) → c_9
  162. Intersect(k, l) → H_2
  163. Intersect(l_1, p_1) → I_2
  164. Segment(H_2, I_2) → d_9
  165. Intersect(m_1, r_1) → J_2
  166. Intersect(e, f) → K_2
  167. Segment(J_2, K_2) → e_9