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Cyclic Quadrilateral Properties Demo

Cyclic Quadrilateral Properties Demo
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2016-08-11 16:00
CirclePolygonAngleTrigonometryGeometric Construction
This example demonstrates the properties of cyclic quadrilaterals, focusing on the Japanese Theorem for Cyclic Quadrilaterals. It involves core concepts like circles, inscribed quadrilaterals, angle bisectors, sine rule, and geometric transformations. The construction utilizes commands such as Circle, Point, Rotate, Segment, Dilate, Vector, Ray, Polygon, AngularBisector, Intersect, OrthogonalLine, Min, CircleSector, Translate, Line, Midpoint, and Angle. It is suitable for high school and above students to understand the geometric properties and applications of cyclic quadrilaterals and related theorems.

GeoGebra Commands & Parameters

  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