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无字证明：a²-b²

无字证明：a²-b²
几何画板及绘图内容加载中，请稍等……
2019-12-29 16:08
向量函数解析几何几何作图
所属专题 Algebraic puzzles for binomial formula
①本范例通过动态图形演示了差平方公式 a²-b² 的几何意义；②涉及核心数学概念包括代数、几何对称性、向量运算；③构造要点：运用了 Circle（圆）、Line（线）、Intersect（交点）、Polygon（多边形）、Midpoint（中点）、Vector（向量）、Rotate（旋转）、Segment（线段）、Tangent（切线）、Point（点）等 GeoGebra 命令与对象；④适合高中数学教学，用于直观理解代数恒等式的几何背景。

GeoGebra 命令及参数

  1. Circle(A, a) → c
  2. Line(A, xAxis) → d
  3. Intersect(c, d) → B, C
  4. Circle(C, a) → e
  5. Line(C, yAxis) → f
  6. Intersect(e, f) → D, E
  7. Line(D, d) → g
  8. Intersect(c, g) → F, G
  9. Polygon(F, D, C, A) → poly1, f_1, d_1, c_1, a_1
  10. Intersect(g, f_1) → H
  11. Circle(F, b) → h
  12. Intersect(h, g) → I, J
  13. Line(J, f) → i
  14. Intersect(h, a_1) → K,
  15. Line(K, g) → j
  16. Intersect(i, j) → L
  17. Polygon(F, J, L, K) → poly2, f_2, j_1, l, k
  18. Circle(F, a / 6) → p
  19. Circle(D, a / 6) → q
  20. Circle(C, a / 6) → r
  21. Circle(A, a / 6) → s
  22. Intersect(p, g) → M, N
  23. Intersect(r, d) → Q, R
  24. Intersect(q, g) → S, T
  25. Intersect(q, f) → U, V
  26. Intersect(r, f) → W, Z
  27. Line(T, yAxis) → m
  28. Line(U, j) → n
  29. Line(M, i) → t_2
  30. Line(Z, d) → a1
  31. Intersect(p, n) → A1, B1
  32. Intersect(s, a1) → C1, D1
  33. Intersect(j, t_2) → E1
  34. Intersect(j, m) → F1
  35. Intersect(i, n) → G1
  36. Midpoint(G1, A1) → H1
  37. Midpoint(G1, U) → I1
  38. Midpoint(T, R) → J1
  39. Midpoint(C1, Z) → K1
  40. Midpoint(F, M) → L1
  41. Midpoint(K, E1) → M1
  42. Vector(J1, T) → u
  43. Intersect(r, m, 1) → P1
  44. Vector(J1, R) → v
  45. Vector(I1, U) → w
  46. Vector(I1, G1) → z
  47. Vector(H1, G1) → u1
  48. Vector(H1, A1) → v1
  49. Midpoint(L1, M1) → Q1
  50. Vector(Q1, L1) → w1
  51. Vector(Q1, M1) → z1
  52. Vector(K1, Z) → d1
  53. Vector(K1, C1) → e1
  54. Midpoint(J, K) → S1
  55. Intersect(j, d_1) → U1
  56. Intersect(i, d) → V1
  57. Midpoint(A, C1) → W1
  58. Intersect(i, a1) → Z1
  59. Midpoint(V1, Z1) → A2
  60. Midpoint(C, Z) → B2
  61. Midpoint(W1, A2) → C2
  62. Midpoint(A2, B2) → D2
  63. Midpoint(D, T) → E2
  64. Midpoint(U1, F1) → F2
  65. Midpoint(E2, F2) → G2
  66. Vector(G2, E2) → h1
  67. Vector(G2, F2) → i1
  68. Vector(C2, A2) → j1
  69. Vector(C2, W1) → k1
  70. Vector(D2, A2) → l1
  71. Vector(D2, B2) → m1
  72. Polygon(J, D, U1, L) → poly3, j_2, d_2, u1_1, l_1
  73. Rotate(J, ((-Animation)) / 10, I2) → J'
  74. Rotate(D, ((-Animation)) / 10, I2) → D'
  75. Rotate(U1, ((-Animation)) / 10, I2) → U1'
  76. Rotate(L, ((-Animation)) / 10, I2) → L'
  77. Polygon(J', D', U1', L') → poly3', j', d', u1', l'
  78. Line(U1', yAxis) → q1
  79. Rotate(L, ((-Animation)) / 10, O) → P
  80. Line(L, yAxis) → c1
  81. Rotate(L, ((-Animation)) / 10, O) → H2
  82. Rotate(K, ((-Animation)) / 10, N1) → O1
  83. Segment(O1, H2) → b1
  84. Circle(H2, b) → f1
  85. Intersect(f1, b1) → J2,
  86. Circle(J2, a - b) → g1
  87. Tangent(J2, f1) → n1, p1
  88. Line(H2, n1) → r1
  89. Intersect(g1, n1) → K2, L2
  90. Line(K2, b1) → s1
  91. Intersect(r1, s1) → M2
  92. Polygon(K2, J2, H2, M2) → poly4, k2, j2, h2, m2
  93. Rotate(U1, ((-Animation)) / 10, R1) → N2
  94. Polygon(L, U1, C, V1) → poly5, l_2, u1_2, c_2, v1_1
  95. Rotate(C, ((-Animation)) / 10, R1) → O2
  96. Segment(O2, N2) → t1
  97. Circle(N2, a - b) → c2
  98. Tangent(O2, c2) → a2, b2
  99. Line(N2, a2) → d2
  100. Circle(O2, a - b) → e2
  101. Intersect(e2, a2) → P2, Q2
  102. Line(Q2, t1) → f2
  103. Intersect(d2, f2) → R2
  104. Polygon(O2, N2, R2, Q2) → poly6, o2, n2, r2, q2
  105. Circle(A, a / 6) → g2
  106. Line(L1, M1) → i2
  107. Line(A, V1) → l2
  108. Intersect(i2, l2) → S2
  109. Midpoint(M1, S2) → U2
  110. Intersect(g2, l2) → V2, W2
  111. Vector(E1, M) → u2
  112. Vector(E1, V2) → v2
  113. Vector(U2, M1) → w2
  114. Vector(U2, S2) → z2
  115. Polygon(F, J, L, K) → poly7, f_3, j_3, l_3, k_1
  116. Polygon(F, D, C, A, K) → poly8, f_4, d_3, c_3, a_2, k_2
  117. Polygon(U1', H2, J', D') → poly9, u1'_1, h2_1, j'_1, d'_1
  118. Polygon(K2, J2, P, M2) → poly10, k2_1, j2_1, p_1, m2_1
  119. Polygon(O2, Q2, R2, N2) → poly11, o2_1, q2_1, r2_1, n2_1
  120. Line(Z2, yAxis) → s2
  121. Point(s2) → B3
  122. Point(s2) → C3
  123. Vector(Z2, B3) → p2
  124. Point(s2) → A3
  125. Vector(Z2, A3) → t2
  126. Line(D3, xAxis) → a3
  127. Point(a3) → E3
  128. Point(a3) → F3
  129. Vector(D3, F3) → u3
  130. Vector(D3, E3) → v3
  131. Line(B3, xAxis) → b3
  132. Circle(B3, a / 6) → c3
  133. Intersect(c3, b3) → G3, H3
  134. Line(G3, p2) → d3
  135. Point(d3) → I3
  136. Point(d3) → J3
  137. Point(d3) → K3
  138. Midpoint(I3, G3) → L3
  139. Midpoint(I3, J3) → M3
  140. Midpoint(J3, K3) → N3
  141. Vector(L3, G3) → w3
  142. Point(d3) → O3
  143. Vector(L3, O3) → z3
  144. Vector(M3, I3) → e3
  145. Vector(M3, J3) → f3
  146. Vector(N3, J3) → g3
  147. Vector(N3, K3) → h3
  148. Polygon(P3, Q3, R3, S3) → poly12, p3, q3, r3, s3
  149. Polygon(U3, V3, W3, Z3) → poly13, u3_1, v3_1, w3_1, z3_1
  150. Segment(A4, B4) → i3
  151. Segment(C4, D4) → j3
  152. Segment(E4, F4) → k3
  153. Segment(G4, H4) → l3

AI 绘图提示语参考

绘制一个几何图形，用于直观展示平方差公式。先画一个大圆，圆心在原点，与横轴交于两点。以右侧交点为圆心，画等半径圆，与纵轴交于两点。连接这些点形成一个大正方形。在大正方形内部，以左下顶点为圆心，画一个半径较小的圆，与底边交于两点。从右侧交点作竖线，从小圆与竖线的交点作横线，与另一条线相交，形成一个小正方形。最后，在大正方形的四个顶点处各画一个小圆，并添加若干辅助线和中点，用箭头表示向量，以展示面积关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。