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圆内接四边形性质演示

圆内接四边形性质演示
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2016-08-11 16:00
圆多边形角三角函数几何作图
①本范例演示圆内接四边形的对角互补性质及日本定理；②涉及核心概念包括圆、内接四边形、角平分线、正弦定理、几何变换；③构造中运用了Circle（圆）、Point（点）、Rotate（旋转）、Segment（线段）、Dilate（缩放）、Vector（向量）、Ray（射线）、Polygon（多边形）、AngularBisector（角平分线）、Intersect（交点）、OrthogonalLine（垂线）、Min（最小值）、CircleSector（扇形）、Translate（平移）、Line（直线）、Midpoint（中点）、Angle（角度）等命令；④适合高中及以上数学学习者，用于理解圆内接四边形的几何性质及定理应用。

GeoGebra 命令及参数

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Rotate(C, 2°, A) → C'
  4. CircleArc(A, C', C) → d
  5. Point(d) → D
  6. CircleArc(A, D, C) → e
  7. Point(e) → E
  8. CircleArc(A, E, C) → f
  9. Point(f) → F
  10. CircleArc(A, F, C) → g
  11. Point(g) → G
  12. Segment(D, E) → b
  13. Segment(E, F) → h
  14. Segment(D, G) → a
  15. Segment(G, F) → i
  16. Circle(A, B) → e_1
  17. Dilate(E, If[0 < j ≤ 1, 1 - j, 0], G) → E'
  18. Vector(E, E') → u
  19. Ray(D, G) → l
  20. Point(l) → P
  21. Dilate(F, If[2 < j ≤ 3, 3 - j, 0], D) → F'
  22. Vector(F, F') → v
  23. Ray(G, D) → n
  24. Point(n) → Q
  25. Polygon(G, D, E, F) → poly1, g_2, d_2, e_2, f_2
  26. Segment(G, E) → m
  27. Segment(D, F) → s
  28. Dilate(F, If[0 < j ≤ 1, 1 - j, 0], D) → F'_1
  29. Segment(F, F'_1) → i_1
  30. AngularBisector(E, F, D) → k
  31. AngularBisector(E, D, F) → p
  32. Intersect(k, p) → H
  33. OrthogonalLine(H, h) → q
  34. Intersect(q, h) → I
  35. AngularBisector(F, D, G) → r
  36. AngularBisector(G, F, D) → t
  37. Intersect(r, t) → J
  38. OrthogonalLine(J, i) → f_1
  39. Intersect(f_1, i) → K
  40. AngularBisector(E, F, G) → g_1
  41. AngularBisector(G, E, F) → h_1
  42. Intersect(g_1, h_1) → L
  43. OrthogonalLine(L, h) → j_1
  44. Intersect(j_1, h) → M
  45. AngularBisector(E, G, D) → k_1
  46. AngularBisector(G, E, D) → l_1
  47. Intersect(k_1, l_1) → N
  48. Rotate(I, If[1 < j ≤ 2, (((j - 1) * 2) * π), (2 * π)], H) → J'
  49. CircleArc(H, I, J') → c_1
  50. Circle(H, I) → d_1
  51. Rotate(K, If[2 < j ≤ 3, (((j - 2) * 2) * π), (2 * π)], J) → K'
  52. CircleArc(J, K, K') → p_1
  53. Circle(J, K) → q_1
  54. Dilate(G, If[3 < j ≤ 4, 4 - j, 0], E) → G'
  55. Segment(G, G') → m_1
  56. Rotate(M, If[4 < j ≤ 5, (((j - 4) * 2) * π), (2 * π)], L) → M'
  57. CircleArc(L, M, M') → r_1
  58. Circle(L, M) → s_1
  59. OrthogonalLine(N, a) → n_1
  60. Intersect(n_1, a) → O
  61. Rotate(O, If[5 < j ≤ 6, (((j - 5) * 2) * π), (2 * π)], N) → O'
  62. CircleArc(N, O, O') → t_1
  63. Circle(N, O) → c_2
  64. Dilate(N, If[6 < j ≤ 7, 7 - j, 0], H) → N'
  65. Segment(N, N') → a_1
  66. Segment(H, N) → b_1
  67. Segment(H, L) → h_2
  68. Min({b_1, h_2}) → o
  69. Circle(H, (0.3 * o)) → k_2
  70. Intersect(k_2, c, 1) → R
  71. Intersect(k_2, c, 1) → S
  72. Intersect(c, h_2) → T,
  73. Intersect(k_2, h_2) → U,
  74. Intersect(k_2, b_1) → V,
  75. CircleSector(H, U, V) → p_2
  76. Dilate(H, If[8 < j ≤ 9, 9 - j, 0], L) → H'
  77. Segment(H, H') → i_2
  78. Dilate(N, If[7 < j ≤ 8, 8 - j, 0], J) → N'_1
  79. Vector(N, N'_1) → w_2
  80. Translate(N, w_2) → N'_2
  81. Translate(N', w_2) → N''
  82. Segment(N'_2, N'') → j_2
  83. Dilate(H, If[9 < j ≤ 10, 10 - j, 0], N) → H'_1
  84. Vector(H, H'_1) → u_1
  85. Translate(H, u_1) → H'_2
  86. Translate(H', u_1) → H''
  87. Segment(H'_2, H'') → l_2
  88. Polygon(J, N, H, L) → poly2, j_3, n_2, h_3, l_3
  89. Dilate(H, If[11 < j ≤ 12, 12 - j, 0], L) → H'_3
  90. Vector(H, H'_3) → v_1
  91. Translate(p_2, v_1) → q_2
  92. Rotate(q_2, If[12 < j ≤ 13, ((j - 12) * π / 2), π / 2], L) → r_2
  93. Dilate(L, If[13 < j ≤ 14, 14 - j, 0], J) → L'
  94. Vector(L, L') → w_1
  95. Translate(r_2, w_1) → s_2
  96. Rotate(s_2, If[14 < j ≤ 15, ((j - 14) * π / 2), π / 2], J) → t_2
  97. Dilate(J, If[15 < j ≤ 16, 16 - j, 0], N) → J'_1
  98. Vector(J, J'_1) → u_2
  99. Translate(t_2, u_2) → c_3
  100. Rotate(c_3, If[16 < j ≤ 17, ((j - 16) * π / 2), π / 2], N) → d_3
  101. Point(d_1) → W
  102. Point(s_1) → Z
  103. Point(q_1) → A_1
  104. Point(c_2) → B_1
  105. Dilate(H, If[0 < w < 1, 1 - w, 0], W) → H'_4
  106. Dilate(L, If[0 < w < 1, 1 - w, 0], Z) → L'_1
  107. Dilate(J, If[0 < w < 1, 1 - w, 0], A_1) → J'_2
  108. Dilate(N, If[0 < w < 1, 1 - w, 0], B_1) → N'_3
  109. Segment(H, W) → m_2
  110. Segment(L, Z) → a_2
  111. Segment(J, A_1) → b_2
  112. Segment(N, B_1) → f_3
  113. Intersect(c, g_3) → D_1, E_1
  114. OrthogonalLine(A, g_3) → i_3
  115. Intersect(c, i_3) → C_1, G_1
  116. Line(C_1, g_3) → k_3
  117. Line(G_1, g_3) → m_3
  118. Midpoint(F_1, E_1) → H_1
  119. Line(H_1, i_3) → n_3
  120. Line(F_1, i_3) → p_3
  121. Line(N, i_3) → t_3
  122. Segment(H, H'_4) → a_3
  123. Segment(L, L'_1) → b_3
  124. Segment(J, J'_2) → e_3
  125. Segment(N, N'_3) → f_4
  126. Circle(H_1, o_1 / 2) → c_4
  127. Circle(F_1, o_1 / 2) → d_4
  128. Intersect(c_4, n_3, 2) → M_1
  129. Intersect(d_4, p_3, 2) → N_1
  130. Intersect(c_4, n_3, 1) → O_1
  131. Intersect(d_4, p_3, 1) → P_1
  132. Segment(O_1, M_1) → g_4
  133. Line(L, t_3) → q_3
  134. Line(J, t_3) → r_3
  135. Line(H, t_3) → s_3
  136. Point(s_3) → I_1
  137. Point(q_3) → J_1
  138. Point(r_3) → K_1
  139. Point(t_3) → Q_1
  140. Angle(B_1, N, Q_1) → α
  141. Angle(W, H, I_1) → β
  142. Angle(Z, L, J_1) → γ
  143. Angle(A_1, J, K_1) → δ
  144. Circle(P_1, f_4) → e_4
  145. Intersect(e_4, p_3, 2) → L_1
  146. Circle(O_1, e_3) → h_4
  147. Intersect(h_4, n_3, 2) → R_1
  148. Dilate(P_1, If[1 < w < 2, 2 - w, 0], L_1) → S_1
  149. Dilate(O_1, If[1 < w < 2, 2 - w, 0], R_1) → T_1
  150. Dilate(L_1, If[2 < w < 3, 3 - w, 0], N_1) → U_1
  151. Dilate(R_1, If[2 < w < 3, 3 - w, 0], M_1) → V_1
  152. Segment(P_1, S_1) → i_4
  153. Segment(S_1, U_1) → j_4
  154. Segment(O_1, T_1) → k_4
  155. Segment(T_1, V_1) → l_4
  156. Dilate(N, If[4 < w < 5, 5 - w, 0], P_1) → N'_4
  157. Vector(N, N'_4) → v_2
  158. Translate(N, v_2) → N'_5
  159. Translate(N'_3, v_2) → W_1
  160. Segment(N'_5, W_1) → m_4
  161. Rotate(N'_5, If[4 < w ≤ 5, ((w - 4) * α), α], N'_5) → Z_1
  162. Rotate(W_1, If[4 < w ≤ 5, ((w - 4) * α), α], N'_5) → A_2
  163. Segment(Z_1, A_2) → n_4
  164. Dilate(L, If[5 < w < 6, 6 - w, 0], S_1) → L'_2
  165. Vector(L, L'_2) → u_3
  166. Translate(L, u_3) → L'_3
  167. Translate(L'_1, u_3) → B_2
  168. Segment(L'_3, B_2) → p_4
  169. Rotate(L'_3, If[5 < w ≤ 6, ((w - 5) * γ), γ], L'_2) → C_2
  170. Rotate(B_2, If[5 < w ≤ 6, ((w - 5) * γ), γ], L'_2) → D_2
  171. Segment(C_2, D_2) → q_4
  172. Dilate(J, If[6 < w < 7, 7 - w, 0], O_1) → J'_3
  173. Vector(J, J'_3) → v_3
  174. Translate(J, v_3) → J'_4
  175. Translate(J'_2, v_3) → E_2
  176. Segment(J'_4, E_2) → r_4
  177. Rotate(J'_4, If[6 < w ≤ 7, ((w - 6) * δ), δ], J'_3) → G_2
  178. Rotate(E_2, If[6 < w ≤ 7, ((w - 6) * δ), δ], J'_3) → H_2
  179. Segment(G_2, H_2) → s_4
  180. Dilate(H, If[7 < w ≤ 8, 8 - w, 0], T_1) → H'_5
  181. Vector(H, H'_5) → w_3
  182. Translate(H, w_3) → H'_6
  183. Translate(H'_4, w_3) → I_2
  184. Segment(H'_6, I_2) → t_4
  185. Rotate(H'_6, If[7 < w ≤ 8, ((w - 7) * β), β], H'_5) → K_2
  186. Rotate(I_2, If[7 < w ≤ 8, ((w - 7) * β), β], H'_5) → L_2
  187. Segment(K_2, L_2) → a_4