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Geometric Proof of a²-b²

Geometric Proof of a²-b²
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2019-12-29 02:41
FunctionCoordinate GeometryGeometric Construction
In collections Algebraic puzzles for binomial formula
This example demonstrates the geometric proof of the difference of squares formula a²-b². Key mathematical concepts include algebra and coordinate geometry. Using objects such as circles, lines, and vectors, it illustrates the geometric interpretation of the difference of squares formula. Suitable for middle school and above students learning algebra and geometry.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a geometric proof of the difference of squares formula a²-b². Start by drawing a circle, then draw a line along the x-axis, and find the intersection points. Use one of these points as the center to draw another circle, which intersects the first circle at two points. Connect these points with the center to form a vector, and translate the center along this vector to create a new point. Rotate this new point to form a series of points, and connect them to form polygons. Continue constructing circles, lines, and polygons to form a complex geometric figure, illustrating the geometric meaning of the difference of squares formula.
The prompt above was compiled from the original drawing commands and is for reference only.