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勾股定理的重组证明

勾股定理的重组证明
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2013-05-30 05:02
测试改名0多边形几何作图
所属专题 Pythagorean Theorem
这个范例演示了勾股定理的著名“重组”证明，通过图形的移动和重组来展示定理。它涉及的核心数学概念是勾股定理和几何变换。作图要点包括使用多边形、中点、射线、旋转、圆、线段、角度、重心、正交线、交点、向量等 GeoGebra 命令和对象。适合初中及以上学生学习和理解几何证明方法。

GeoGebra 命令及参数

  1. Polygon(A, C, B) → poly1, b_1, a_1, c_2
  2. Midpoint(b_1) → M
  3. Ray(B, M) → n_2
  4. Rotate(B, (-(((90 * Progress_4))°)), B) → B'
  5. Circle(B', sqrt(b^(2) + ((0.5 * a))^(2)) - 0.1) → d_3
  6. Rotate(L, ((90 * Progress_2))°, J) → L'
  7. Rotate(L', ((Progress_3 * 90))°, L') → L''
  8. Circle(L'', sqrt(a^(2) + ((0.5 * b))^(2)) - 0.1) → c_3
  9. Rotate(K, ((90 * Progress_2))°, J) → K'
  10. Polygon(J, K', L') → poly4, l', j, k'
  11. Midpoint(l') → Z
  12. Ray(L', Z) → m_2
  13. Rotate(E, ((90 * Progress))°, D) → E'
  14. Rotate(F, ((90 * Progress))°, D) → F'
  15. Polygon(F', D, E') → poly2, e', f', d
  16. Rotate(H, ((90 * Progress_1))°, G) → H'
  17. Rotate(I, ((90 * Progress_1))°, G) → I'
  18. Polygon(H', I', G) → poly3, g, h', i'
  19. If(Progress ≟ 0, 0, 1) → e
  20. If(Progress_1 ≟ 0, false, true) → h
  21. Rotate(J, ((Progress_3 * 90))°, L') → J'
  22. Rotate(K', ((Progress_3 * 90))°, L') → K''
  23. Polygon(J', K'', L'') → poly4', l'', j', k''
  24. If(Progress ≟ 1, If[Progress_1 ≟ 0, true, false], true) → Progress_{visibility}
  25. If(Progress ≟ 1, If[Progress_2 ≟ 0, true, false], false) → Progress1_{visibility}
  26. Polygon(C, A, B) → poly5, b_2, c_1, a_2
  27. Rotate(C, (-(((90 * Progress_4))°)), B) → C'
  28. Rotate(A, (-(((90 * Progress_4))°)), B) → A'
  29. Polygon(C', A', B') → poly5', b', c', a'
  30. If(Progress_2 ≟ 0, false, If[Progress_3 ≟ 0, true, false]) → i
  31. If(Progress_3 ≟ 0, true, false) → k
  32. If(Progress_1 ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Progress2_{visibility}
  33. If(Progress_3 ≟ 0, false, true) → l
  34. Segment(J, A) → m
  35. Segment(A, D) → n
  36. Segment(D, G) → p
  37. Segment(G, J) → q
  38. If(Progress_2 ≟ 1, true, false) → square
  39. Angle(B, A, C) → α
  40. Angle(E', D, F') → β
  41. Angle(H', G, I') → γ
  42. Angle(K', J, L') → δ
  43. If(Progress ≟ 1, true, false) → Right1
  44. If(Progress_1 ≟ 1, true, false) → Right2
  45. Angle(E', F', C) → ε
  46. Angle(H', I', F') → ζ
  47. Angle(C, H', I') → η
  48. Angle(B, C, H') → θ
  49. If(Progress ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Blue1
  50. If(Progress_1 ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Blue2
  51. If(Progress_2 ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Blue3
  52. Polygon(C, L', E', B) → poly6, r, s, t, d_1
  53. If(Progress_2 ≟ 1 ∧ Progress_3 ≟ 0, true, false) → Csquare
  54. Polygon(J, L, J', L') → poly7, j_1, l_1, j'_1, l'_1
  55. Polygon(L, A, B, A') → poly8, l_2, a_3, b_3, a'_1
  56. Angle(J, L', J') → ι
  57. Angle(L', J', L) → κ
  58. Angle(J', L, J) → λ
  59. Angle(L, A', B) → μ
  60. Angle(A', B, A) → ν
  61. Angle(A, L, A') → ξ
  62. If(Progress_3 ≟ 1, true, false) → Bluesquares
  63. Midpoint(c_2) → N
  64. Midpoint(a_1) → O
  65. Midpoint(e') → P
  66. Midpoint(f') → Q
  67. Midpoint(d) → R
  68. Midpoint(h') → S
  69. Midpoint(i') → T
  70. Midpoint(g) → U
  71. If(Progress_1 ≟ 1, true, false) → Legs_{3rd}
  72. Midpoint(k') → V
  73. Midpoint(j) → W
  74. If(Progress ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Hypo_{2nd}
  75. If(Progress_1 ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Hypo_{3rd}
  76. If(Progress_2 ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Hypo_{4th}
  77. Centroid(poly6) → A_1
  78. If(Progress ≟ 1, true, false) → Legs_1
  79. If(Progress_2 ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Legs_{4th}
  80. Midpoint(l'_1) → B_1
  81. Midpoint(k'') → C_1
  82. Midpoint(j_1) → D_1
  83. Midpoint(l_1) → E_1
  84. Midpoint(a'_1) → F_1
  85. Midpoint(b_3) → G_1
  86. Midpoint(a_3) → H_1
  87. Midpoint(l_2) → I_1
  88. If(Progress_3 ≟ 1, true, false) → Bluelabels
  89. If(Progress_2 ≟ 1 ∧ Progress_3 ≟ 0, true, false) → textofcsquared
  90. Centroid(poly7) → J_1
  91. Centroid(poly8) → K_1
  92. If(Progress_3 ≟ 1, true, false) → aandbsquareds
  93. Line(L_1, yAxis) → c
  94. Line(M_1, yAxis) → e_1
  95. Point(c) → N_1
  96. OrthogonalLine(N_1, c) → f_1
  97. Point(c) → O_1
  98. OrthogonalLine(O_1, c) → g_1
  99. Point(c) → P_1
  100. OrthogonalLine(P_1, c) → h_1
  101. Intersect(e_1, f_1) → Q_1
  102. Intersect(e_1, g_1) → R_1
  103. Intersect(e_1, h_1) → S_1
  104. Polygon(P_1, O_1, R_1, S_1) → poly9, p_1, o_1, r_1, s_1
  105. Polygon(O_1, N_1, Q_1, R_1) → poly10, o_2, n_1, q_1, r_2
  106. If(Progress ≟ 0, true, false) → Title
  107. Segment(T_1, U_1) → i_1
  108. Segment(U_1, V_1) → k_1
  109. Segment(V_1, W_1) → m_1
  110. Segment(W_1, T_1) → t_1
  111. If(Progress ≟ 0, true, false) → o
  112. Vector(Z_1, A_2) → u
  113. Line(A_2, u) → d_2
  114. Point(d_2) → E_2
  115. Point(d_2) → F_2
  116. Vector(E_2, F_2) → z
  117. If(Tutorial ≟ true, If[Progress ≟ 0, true, false], false) → tutorial_{1st(2)}
  118. If(Tutorial ≟ true, If[Progress ≟ 0 ∨ Progress ≟ 1, false, true], false) → tutorial_{1stto2nd}
  119. If(Tutorial ≟ true, If[Progress ≟ 0, true, false], false) → tutorial_{1st}
  120. If(Tutorial ≟ true, If[Progress ≟ 1 ∧ Progress_1 ≟ 0, true, false], false) → Tutorial_{2nd}
  121. If(Tutorial ≟ true, If[Progress ≟ 1, false, true], false) → Vector_1
  122. Angle(K'', J', L'') → ο
  123. Angle(B', A', C') → ρ
  124. Ray(G, J) → e_2
  125. Ray(D, A) → f_2
  126. OrthogonalLine(G_2, f_2) → g_2
  127. Intersect(g_2, f_2) → H_2
  128. Segment(G_2, J) → h_2
  129. Segment(H_2, A) → i_2
  130. Midpoint(G_2, H_2) → I_2
  131. Vector(I_2, G_2) → z_1
  132. Vector(I_2, H_2) → u_2
  133. If(Progress_2 ≟ 1, true, false) → aplusb
  134. Circle(L'', (0.5 * a)) → k_2
  135. Point(k_2) → J_2
  136. Segment(L'', J_2) → j_2
  137. Rotate(J_2, ((90 * Progress_3))°, L'') → K_2
  138. Circle(B', (0.5 * b)) → p_2
  139. Point(p_2) → L_2
  140. Rotate(L_2, (-(((90 * Progress_3))°)), B') → M_2
  141. If(Progress_3 ≟ 0, false, true) → Stillaandb
  142. If(Progress_3 ≟ 0 ∨ Progress_3 ≟ 1, false, true) → movingaandb
  143. If(Progress_3 ≟ 0, false, true) → movingrightangles
  144. Circle(L'', sqrt(a^(2) + ((0.5 * b))^(2))) → q_2
  145. Circle(B', sqrt(b^(2) + ((0.5 * a))^(2))) → s_2
  146. Point(q_2) → N_2
  147. Point(s_2) → O_2
  148. Rotate(N_2, ((90 * Progress_3))°, L'') → P_2
  149. Rotate(O_2, (-(((90 * Progress_3))°)), B') → Q_2
  150. Intersect(c_3, m_2) → R_2,
  151. Rotate(R_2, ((90 * Progress_3))°, L'') → T_2
  152. Intersect(d_3, n_2) → S_2,
  153. Angle(J, L'', J') → σ
  154. Angle(A', B', A) → τ
  155. If(Progress_4 ≟ 1, true, false) → SecondBlueSquare
  156. If(Progress_3 ≟ 1, true, false) → Progress4_{visibility}
  157. If(Progress_2 ≟ 1, If[Progress_4 ≟ 0, true, false], false) → Progress3_{visibility}
  158. If(Progress_4 ≟ 0, true, false) → Hypo_{1st}
  159. If(Progress_4 ≟ 0, true, false) → Legs_{1st}
  160. If(Progress_2 ≟ 1, If[Progress_3 ≟ 1, false, true], false) → fakecsquared
  161. If(Progress_3 ≟ 1, If[Progress_4 ≟ 1, false, true], false) → SecondEquation
  162. Vector(U_2, V_2) → v_2
  163. If(Progress ≟ 0, false, true) → f
  164. If(Tutorial ≟ true, If[Progress_1 ≟ 1, false, true], false) → AnnoyingTutorial

AI 绘图提示语参考

画一个直角三角形，通过移动和重新排列其部分面积来证明勾股定理。首先展示一个由三角形三边构成的图形，然后将其一部分旋转并平移，与另一部分组合形成一个新的图形，同时用圆和线段辅助展示面积关系。整个过程动态展示形状的变换和重合，最终证明勾股定理的几何意义。
以上提示语为根据原始绘图命令整理而来，仅供参考。