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差平方的几何证明

差平方的几何证明
几何画板及绘图内容加载中，请稍等……
2019-12-29 02:41
函数解析几何几何作图
所属专题 Algebraic puzzles for binomial formula
这个范例通过几何图形展示了差平方公式a²-b²的证明过程。核心数学概念包括代数和解析几何。通过构造圆、线、向量等对象，演示了差平方公式的几何意义。适合初中及以上学生学习代数和几何。

GeoGebra 命令及参数

  1. Circle(A, If[Animation < o_2, Animation, o_2]) → f_1
  2. Line(A, xAxis) → g_1
  3. Intersect(f_1, g_1) → I_1, J_1
  4. Circle(J_1, If[Animation < o_2, Animation / 3, o_2 / 3]) → h_1
  5. Intersect(h_1, f_1) → K_1, L_1, ,
  6. Vector(A, K_1) → j_1
  7. Circle(A, Zoom) → c
  8. Point(c) → B
  9. Translate(B, j_1) → B'_2
  10. Rotate(B, 90°, A) → B'
  11. Rotate(B', 90°, A) → B''
  12. Rotate(B'', 90°, A) → B'''
  13. Translate(B''', j_1) → O_1
  14. Rotate(O_1, If[Animation < 9, 0, If[Animation ≥ 12.14, 3.1400000000000006, Animation - 9]], B'_2) → P_1
  15. Polygon(B, B', B'', B''') → poly1, b, b', b'', b'''
  16. Point(b') → C
  17. Circle(B', C) → d
  18. Intersect(d, b) → D,
  19. Translate(D, j_1) → D'
  20. Line(D, b') → e
  21. Line(C, b) → f
  22. Intersect(e, f) → E
  23. Translate(E, j_1) → E'
  24. Polygon(E', D', B'_2, O_1) → poly3', e', d', b'_3, o_1
  25. Midpoint(d') → T_3
  26. Line(T_3, b''') → a
  27. Mirror(O_1, a) → U_3
  28. Vector(O_1, U_3) → t
  29. Translate(B'_2, Vector[(If[Animation > 14, 1, Animation - 13] * t)]) → A_4
  30. Translate(O_1, Vector[(If[Animation > 14, 1, Animation - 13] * t)]) → V_3
  31. Intersect(o_1, a) → G_4
  32. Rotate(E', If[Animation < 9, 0, If[Animation < 12.14, Animation - 9, 3.1400000000000006]], D') → E''
  33. Polygon(D', E'', P_1, B'_2) → poly5, d'_2, e'', p_1, b'_4
  34. Intersect(a, e'') → B_4
  35. Vector(G_4, B_4) → w_4
  36. Translate(A_4, Vector[(If[Animation > 16, 1, (Animation - 14) / 2] * w_4)]) → E_4
  37. Translate(V_3, Vector[(If[Animation > 16, 1, (Animation - 14) / 2] * w_4)]) → C_4
  38. Midpoint(J_4, K_4) → L_4
  39. Polygon(B'', C, E, B''') → poly2, b''_1, c_1, e_1, b'''_1
  40. Polygon(E, D, B, B''') → poly3, e_2, d_1, b_1, b'''_2
  41. Polygon(C, B', D, E) → poly4, c_2, b'_1, d_2, e_3
  42. Line(B'', f) → g
  43. Circle(C, Zoom / 8) → h
  44. Circle(B, Zoom / 8) → k
  45. Circle(B'', Zoom / 8) → p
  46. Circle(B''', Zoom / 8) → q
  47. Line(B, f) → i
  48. Intersect(k, i) → F, G
  49. Intersect(q, g) → H, I
  50. Line(B''', b') → j
  51. Line(B'', B') → l
  52. Circle(B', Zoom / 8) → r
  53. Intersect(r, l) → J, K
  54. Intersect(k, j) → L, M
  55. Intersect(q, j) → N, O
  56. Intersect(g, l) → P
  57. Intersect(p, l) → Q, R
  58. Intersect(p, g) → S, T
  59. Intersect(h, f) → U, V
  60. Circle(D, Zoom / 8) → s
  61. Intersect(s, e) → W, Z
  62. Intersect(r, i) → A_1, B_1
  63. Midpoint(Q, O) → C_1
  64. Midpoint(H, F) → D_1
  65. Midpoint(L, W) → E_1
  66. Midpoint(W, K) → F_1
  67. Midpoint(B_1, V) → G_1
  68. Midpoint(V, T) → H_1
  69. Vector(H_1, V) → u
  70. Vector(G_1, B_1) → w
  71. Vector(G_1, V) → z
  72. Vector(F_1, K) → m
  73. Vector(D_1, F) → v_1
  74. Vector(E_1, L) → u_1
  75. Vector(D_1, H) → w_1
  76. Vector(C_1, Q) → z_1
  77. Vector(H_1, T) → v
  78. Vector(F_1, W) → n
  79. Vector(E_1, W) → t_2
  80. Vector(C_1, O) → a_1
  81. Vector(A, L_1) → i_1
  82. Translate(B, i_1) → B'_1
  83. Translate(B', i_1) → B''_1
  84. Translate(B'', i_1) → B'''_1
  85. Translate(B''', i_1) → M_1
  86. Polygon(B'_1, B''_1, B'''_1, M_1) → poly1', b'_2, b''_2, b'''_3, m_1
  87. Circle(A, If[Animation < o_2 + 4.5, Animation, o_2 + 4.5]) → k_1
  88. Midpoint(L_1, K_1) → S_1
  89. Midpoint(L_1, S_1) → T_1
  90. Line(A, T_1) → a_2
  91. Intersect(k_1, a_2) → U_1, V_1
  92. Vector(E, V_1) → r_1
  93. Translate(C, r_1) → C'
  94. Translate(B', r_1) → B''_2
  95. Translate(D, r_1) → D'_1
  96. Translate(E, r_1) → E'_1
  97. Polygon(C', B''_2, D'_1, E'_1) → poly4', c', b''_3, d'_1, e'_1
  98. Circle(B''_1, Zoom / 8) → q_1
  99. Circle(B'''_1, Zoom / 8) → s_1
  100. Circle(B''_2, Zoom / 8) → t_1
  101. Circle(C', Zoom / 8) → f_2
  102. Midpoint(B''_1, B'_1) → R_1
  103. Midpoint(B''_2, E'_1) → W_1
  104. Line(R_1, g_1) → l_1
  105. Intersect(l_1, e'_1) → N_1
  106. Midpoint(R_1, N_1) → Z_1
  107. Circle(E'', Zoom / 8) → g_2
  108. Circle(E', Zoom / 8) → h_2
  109. Circle(D', Zoom / 8) → k_2
  110. Circle(O_1, Zoom / 8) → p_2
  111. Circle(P_1, Zoom / 8) → q_2
  112. Circle(B'_2, Zoom / 8) → r_2
  113. Circle(M_1, Zoom / 8) → s_2
  114. Circle(D'_1, Zoom / 8) → c_3
  115. Line(M_1, B'''_1) → n_1
  116. Line(B''_1, B'_1) → b_2
  117. Intersect(s_1, n_1) → A_2, B_2
  118. Intersect(q_1, b_2) → C_2, D_2
  119. Midpoint(B_2, C_2) → E_2
  120. Line(B'''_1, l_1) → i_2
  121. Line(M_1, g_1) → j_2
  122. Line(E', n_1) → l_2
  123. Line(E'', b_2) → m_2
  124. Line(P_1, m_2) → n_2
  125. Line(O_1, n_1) → a_3
  126. Line(D', b_2) → b_3
  127. Vector(E_2, C_2) → u_2
  128. Vector(E_2, B_2) → v_2
  129. Intersect(s_1, i_2) → F_2, G_2
  130. Intersect(s_2, j_2) → H_2, I_2
  131. Line(B''_2, n_2) → d_3
  132. Line(C', m_2) → f_3
  133. Intersect(f_2, f_3) → J_2, K_2
  134. Intersect(t_1, d_3) → L_2, M_2
  135. Line(D'_1, l_1) → g_3
  136. Line(B''_2, l_1) → h_3
  137. Intersect(t_1, h_3) → N_2, O_2
  138. Intersect(c_3, g_3) → P_2, Q_2
  139. Intersect(g_2, m_2) → R_2, S_2
  140. Intersect(h_2, l_2) → T_2, U_2
  141. Intersect(p_2, a_3) → V_2, W_2
  142. Intersect(r_2, b_3) → Z_2, A_3
  143. Intersect(k_2, b_3) → B_3, C_3
  144. Intersect(q_2, n_2) → D_3, E_3
  145. Intersect(k_2, d'_2) → F_3,
  146. Intersect(r_2, p_1) → G_3,
  147. Midpoint(J_2, L_2) → J_3
  148. Midpoint(O_2, Q_2) → K_3
  149. Midpoint(F_2, H_2) → H_3
  150. Vector(H_3, F_2) → w_2
  151. Vector(H_3, H_2) → z_2
  152. Midpoint(U_2, B_3) → I_3
  153. Midpoint(B_3, R_2) → L_3
  154. Midpoint(V_2, A_3) → M_3
  155. Midpoint(A_3, E_3) → N_3
  156. Midpoint(G_3, F_3) → O_3
  157. Vector(N_3, E_3) → w_3
  158. Vector(N_3, A_3) → z_3
  159. Vector(L_3, R_2) → i_3
  160. Vector(L_3, B_3) → j_3
  161. Vector(O_3, F_3) → m_3
  162. Vector(O_3, G_3) → n_3
  163. Vector(J_3, L_2) → p_3
  164. Vector(J_3, J_2) → q_3
  165. Vector(K_3, O_2) → r_3
  166. Vector(L_4, J_4) → l_3
  167. Vector(K_3, Q_2) → s_3
  168. Midpoint(R_1, Z_1) → P_3
  169. Midpoint(Z_1, N_1) → Q_3
  170. Midpoint(P_3, Z_1) → R_3
  171. Midpoint(Z_1, Q_3) → S_3
  172. Segment(R_3, S_3) → t_3
  173. Mirror(E', a) → E''_1
  174. Vector(E''_1, E') → u_4
  175. Vector(B'_2, D') → v_4
  176. Translate(E', Vector[((-If[Animation > 14, 1, Animation - 13]) * u_4)]) → E''_2
  177. Translate(D', Vector[((-If[Animation > 14, 1, Animation - 13]) * v_4)]) → D''
  178. Point(v_4) → W_3
  179. Translate(E''_1, Vector[(If[Animation > 14, 1, Animation - 13] * t)]) → Z_3
  180. Polygon(V_3, E''_2, D'', A_4, Z_3) → poly6, v_5, e''_2, d'', a_4, z_4
  181. Segment(E', E''_1) → b_4
  182. Translate(E''_2, Vector[(If[Animation > 16, 1, (Animation - 14) / 2] * w_4)]) → D_4
  183. Translate(D'', Vector[(If[Animation > 16, 1, (Animation - 14) / 2] * w_4)]) → D'''
  184. Translate(Z_3, Vector[(If[Animation > 16, 1, (Animation - 14) / 2] * w_4)]) → F_4
  185. Polygon(C_4, D_4, D''', E_4, F_4) → poly6', c_4, d_4, d''', e_4, f_4
  186. Point(a_4) → H_4
  187. Point(a_4) → I_4
  188. Segment(E_4, D''') → g_4
  189. Segment(D''', D_4) → h_4
  190. Segment(C_4, E_4) → i_4
  191. Vector(L_4, K_4) → k_3
  192. Midpoint(O_4, N_4) → P_4
  193. Vector(P_4, O_4) → v_3
  194. Vector(P_4, N_4) → u_3
  195. Vector(P_4, N_4) → j_4
  196. Midpoint(d''') → Q_4

AI 绘图提示语参考

绘制一个几何证明，展示差平方公式a²-b²。首先画一个圆，然后画一条x轴线，圆与x轴相交得到两个点。接着以其中一个交点为圆心画一个圆，与第一个圆相交得到两个点。连接这两个点与圆心形成向量，以此向量为方向平移圆心得到新点。旋转新点形成一系列点，连接这些点形成多边形。继续构造一系列圆、线和多边形，最终形成复杂的几何图形，展示差平方公式的几何意义。
以上提示语为根据原始绘图命令整理而来，仅供参考。