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Pythagoras' Theorem --Area Dissection

Pythagoras' Theorem --Area Dissection
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2017-12-11 15:54
TriangleCoordinate GeometryGeometric Construction
This example demonstrates a method of proving Pythagoras' Theorem through area dissection. Key mathematical concepts include area and the Pythagorean Theorem. The construction involves midpoints, circles, line segments, rotations, and intersections, illustrating the geometric construction process. Suitable for middle school and above students learning geometric proofs.

GeoGebra Commands & Parameters

  1. Midpoint(A, C) → A_0
  2. Circle(A_0, C) → c
  3. Point(c) → B
  4. Segment(A, C) → a
  5. Segment(C, B) → b
  6. Segment(B, A) → d
  7. Rotate(B, 90°, C) → B_2
  8. Rotate(C, -90°, B) → B_1
  9. Rotate(B, -90°, A) → A_2
  10. Rotate(A, 90°, B) → A_1
  11. Rotate(A, -90°, C) → C_2
  12. Rotate(C, 90°, A) → C_1
  13. Segment(A, A_2) → e
  14. Segment(A_2, A_1) → f
  15. Segment(A_1, B) → g
  16. Segment(B, B_1) → h
  17. Segment(B_1, B_2) → i
  18. Segment(B_2, C) → j
  19. Segment(A, C_1) → k
  20. Segment(C_1, C_2) → l
  21. Segment(C_2, C) → m
  22. Midpoint(A, B_1) → D
  23. Line(D, a) → n
  24. Intersect(n, j) → E
  25. Midpoint(D, E) → F
  26. OrthogonalLine(F, n) → p_4
  27. Intersect(p_4, b) → G
  28. Intersect(p_4, i) → H
  29. Segment(D, E) → q_3
  30. Segment(H, G) → r_3
  31. Midpoint(C, A_1) → I
  32. Line(I, a) → s_3
  33. Intersect(s_3, e) → J
  34. Midpoint(J, I) → K
  35. OrthogonalLine(K, s_3) → t_3
  36. Segment(J, I) → a_1
  37. Intersect(t_3, f) → L
  38. Intersect(t_3, d) → M
  39. Segment(L, M) → b_1
  40. Distance(A, B) → AB
  41. Distance(B, C) → BC
  42. Min(AB, BC) → MinS
  43. If(BC > AB, Distance[D, F], Distance[L, K]) → Len2
  44. Circle(C, Len2) → c_1
  45. Circle(C_1, Len2) → d_1
  46. Intersect(c_1, m, 1) → N_1
  47. Intersect(c_1, a, 1) → N_2
  48. Intersect(d_1, l, 1) → N_4
  49. Intersect(d_1, k, 1) → N_3
  50. Line(N_1, b) → e_1
  51. Line(N_2, d) → f_1
  52. Line(N_4, d) → g_1
  53. Line(N_3, b) → h_1
  54. Intersect(e_1, f_1) → O_1
  55. Intersect(e_1, g_1) → O_2
  56. Intersect(g_1, h_1) → O_3
  57. Intersect(f_1, h_1) → O_4
  58. Segment(N_1, O_1) → i_1
  59. Segment(N_2, O_4) → j_1
  60. Segment(N_3, O_3) → k_1
  61. Segment(N_4, O_2) → l_1
  62. Circle(C, Len2) → p_1
  63. Circle(C_1, Len2) → q_1
  64. Intersect(q_1, k, 1) → M_4
  65. Intersect(q_1, l, 1) → M_3
  66. Intersect(p_1, m, 1) → M_2
  67. Intersect(p_1, a, 1) → M_1
  68. Line(M_1, b) → m_1
  69. Line(M_3, b) → n_1
  70. Line(M_4, h) → r_1
  71. Line(M_2, j) → s_1
  72. Intersect(n_1, r_1) → P_4
  73. Intersect(n_1, s_1) → P_3
  74. Intersect(m_1, s_1) → P_2
  75. Intersect(m_1, r_1) → P_1
  76. Segment(M_3, P_4) → t_1
  77. Segment(M_4, P_1) → a_2
  78. Segment(M_1, P_2) → b_2
  79. Segment(M_2, P_3) → c_2
  80. Circle(Q_1, Len2) → d_2
  81. Line(Q_1, m) → e_2
  82. Line(Q_1, a) → f_2
  83. Intersect(d_2, f_2, 1) → Q_2
  84. Intersect(d_2, e_2, 1) → Q_3
  85. Line(Q_3, b) → g_2
  86. Line(Q_2, b) → h_2
  87. Line(Q_2, d) → i_2
  88. Line(Q_3, d) → j_2
  89. Intersect(g_2, i_2) → Q_4
  90. Intersect(h_2, j_2) → Q_5
  91. If(BC > AB, Polygon[Q_1, Q_3, Q_4, Q_2], Polygon[Q_1, Q_2, Q_5, Q_3]) → RedPoly
  92. Circle(R_1, Len2) → k_2
  93. Line(R_1, a) → l_2
  94. Line(R_1, m) → m_2
  95. Intersect(k_2, l_2, 1) → R_2
  96. Intersect(k_2, m_2, 2) → R_3
  97. Line(R_2, d) → n_2
  98. Line(R_3, d) → p_2
  99. Line(R_2, b) → q_2
  100. Line(R_3, b) → r_2
  101. Intersect(p_2, q_2) → R_4
  102. Intersect(n_2, r_2) → R_5
  103. If(BC > AB, (Polygon[R_1, R_2, R_5, R_3] * Polygon[R_1, R_3, R_4, R_5])) → o
  104. If(BC > AB, Polygon[R_1, R_2, R_5, R_3], Polygon[R_1, R_3, R_4, R_2]) → BluePoly
  105. Circle(S_1, Len2) → s_2
  106. Line(S_1, a) → t_2
  107. Line(S_1, m) → a_3
  108. Intersect(s_2, a_3, 2) → S_2
  109. Intersect(s_2, t_2, 2) → S_3
  110. Line(S_2, b) → b_3
  111. Line(S_3, b) → c_3
  112. Line(S_2, d) → d_3
  113. Line(S_3, d) → e_3
  114. Intersect(c_3, d_3) → S_5
  115. Intersect(b_3, e_3) → S_4
  116. If(BC > AB, Polygon[S_1, S_2, S_4, S_3], Polygon[S_1, S_3, S_5, S_2]) → AmberPoly
  117. Circle(T_1, Len2) → f_3
  118. Line(T_1, a) → g_3
  119. Line(T_1, m) → h_3
  120. Intersect(f_3, h_3, 1) → T_2
  121. Intersect(f_3, g_3, 2) → T_3
  122. Line(T_2, d) → i_3
  123. Line(T_3, d) → j_3
  124. Line(T_3, b) → k_3
  125. Line(T_2, b) → l_3
  126. Intersect(j_3, l_3) → T_4
  127. Intersect(i_3, k_3) → T_5
  128. If(BC > AB, Polygon[T_1, T_3, T_4, T_2], Polygon[T_1, T_2, T_5, T_3]) → GreenPoly
  129. Circle(U_1, MinS) → p_3
  130. Line(U_1, g) → m_3
  131. Intersect(p_3, m_3, 2) → U_2
  132. Rotate(U_2, 90°, U_1) → U_3
  133. Rotate(U_1, -90°, U_2) → U_4
  134. Polygon(U_3, U_1, U_2, U_4) → poly1, u_3, u_1, u_2, u_4
  135. Vector(Q_1, C) → p
  136. Vector(R_1, C_2) → q
  137. Vector(S_1, C_1) → r
  138. Vector(T_1, A) → s
  139. Translate(RedPoly, p_t) → RedPoly'
  140. Translate(BluePoly, q_t) → BluePoly'
  141. Translate(AmberPoly, r_t) → AmberPoly'
  142. Translate(GreenPoly, s_t) → GreenPoly'
  143. Vector(U_1, O_1) → u
  144. Vector(U_1, P_1) → v
  145. If(BC > AB, Vector[(Brown * u)], Vector[(Brown * v)]) → u_t
  146. Translate(U_3, u_t) → N
  147. Translate(U_1, u_t) → O
  148. Translate(U_2, u_t) → P
  149. Translate(U_4, u_t) → Q
  150. Polygon(N, O, P, Q) → poly1', n_3, o_1, p_5, q_4
  151. Polygon(A, B, C) → poly2, c_4, a_4, b_4
  152. Circle(B, S90) → d_4
  153. Intersect(d_4, b, 1) → R
  154. Intersect(d_4, d, 1) → S
  155. Rotate(B, -90°, R) → B'
  156. Segment(S, B') → e_4
  157. Segment(B', R) → f_4
  158. Translate(RedPoly, v_1) → RedPoly'_1
  159. Translate(BluePoly, v_2) → BluePoly'_1
  160. Translate(AmberPoly, v_3) → AmberPoly'_1
  161. Translate(GreenPoly, v_4) → GreenPoly'_1
  162. If(BC > AB, Vector[(Rearrange * u)], Vector[(Rearrange * v)]) → v_t
  163. Translate(U_3, v_t) → T
  164. Translate(U_1, v_t) → U
  165. Translate(U_2, v_t) → V
  166. Translate(U_4, v_t) → W
  167. Polygon(T, U, V, W) → poly1'_1, t_4, u_5, v_5, w_1
  168. SelectedIndex(Bahasa) → nn

AI Drawing Prompt Reference

Draw a right triangle ABC with A as the right angle vertex, B as the hypotenuse vertex, and C as the base vertex. Use midpoints, circles, rotations, and intersections to demonstrate the area dissection proof of Pythagoras' Theorem. Start by drawing points A and C, then draw line segment AC. Construct the midpoint A0 of AC and draw circle c with A0 as the center, which intersects AC at point B. Next, draw line segments AB, BC, and CA. Use rotations and intersections to construct a series of auxiliary line segments and points, ultimately proving Pythagoras' Theorem through these auxiliary elements.
The prompt above was compiled from the original drawing commands and is for reference only.