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Open Middle Inscribed Angle Exercise (2)

Open Middle Inscribed Angle Exercise (2)
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2023-05-19 21:58
CircleAngleGeometric Construction
This example demonstrates the relationship between inscribed angles and central angles in a circle. It uses geometric constructions to illustrate angle properties. Key concepts include circle theorems, angle relationships, and geometric construction techniques. The construction involves GeoGebra commands such as Polygon, Circle, Intersect, and others. Suitable for middle or high school mathematics education to understand angle relationships in geometric figures.

GeoGebra Commands & Parameters

  1. Polygon(A0, B, 4) → poly0, f, g, h, i, A, C
  2. Polygon(A3, H, 4) → poly3, j, k, l, m, E, G
  3. Polygon(A7, P, 4) → poly7, n, p, q, r, I, K
  4. Polygon(A1, D, 4) → poly1, s, t, a, b, M, O
  5. Polygon(A6, N, 4) → poly6, c, d, e, f_1, Q, S
  6. Polygon(A2, H2, 4) → poly2, k_1, l_1, m_1, n_1, W, Z
  7. Polygon(A8, A8 + (2, 0), 4) → poly8, g_1, h_1, i_1, j_1, U, V
  8. Polygon(A9, T, 4) → poly9, p_1, q_1, r_1, s_1, A_1, B_1
  9. Polygon(A4, J, 4) → poly4, t_1, a_1, b_1, c_1, C_1, D_1
  10. Polygon(A5, L, 4) → poly5, d_1, e_1, f_2, g_2, E_1, F_1
  11. Polygon(K_1, L_1, 4) → poly10, l_2, m_2, n_2, p_2, M_1, N_1
  12. Polygon(O_1, P_1, 4) → poly12, q_2, r_2, s_2, t_2, Q_1, R_1
  13. Polygon(S_1, T_1, 4) → poly13, a_2, b_2, c_2, d_2, U_1, V_1
  14. Polygon(C_2, D_2, 4) → poly15, i_3, j_3, k_3, l_3, E_2, F_2
  15. Polygon(G_2, H_2, 4) → poly16, m_3, n_3, p_3, q_3, I_2, J_2
  16. Polygon(G_{3}, H_{3}, 4) → poly17, m_{4}, n_{4}, p_{4}, q_{4}, I_{3}, J_{3}
  17. Polygon(C_{3}, D_{3}, 4) → poly14, i_{4}, j_{4}, k_{4}, l_{4}, E_{3}, F_{3}
  18. If(aleft > aright, "Left side is bigger.", aleft < aright, "Right side is bigger", "Both sides are equal!") → text9
  19. nCr(4, 2) → w
  20. If(o_1 ≟ 0, p15 / p10, ((-p15)) / p10) → u_2
  21. If(u_1 ≟ 0, p12 / p13, ((-p12)) / p13) → v_2
  22. If(w_1 ≟ 0, p17 / p14, ((-p17)) / p14) → w_2
  23. If(o_2 ≟ 0, p16, (-p16)) → b_3
  24. Polygon(T_2, U_2, 4) → poly11, s_3, t_3, a_3, c_3, V_2, W_2
  25. Polygon(Z_2, A_3, 4) → poly18, d_3, e_3, f_4, g_4, B_3, M_3
  26. Circle(N_3, O_3) → c_4
  27. Point(c_4) → P_3
  28. Point(c_4) → Q_3
  29. Point(c_4) → R_3
  30. Point(c_4) → S_3
  31. Circle(A_2, B_2) → e_2
  32. Point(e_2) → O_2
  33. Point(e_2) → P_2
  34. Rotate(P_2, (((100 * p13) + (10 * p12) + p16) * pi / 180), O_2) → R_2
  35. Ray(O_2, R_2) → i_2
  36. Intersect(e_2, i_2, 2) → S_2
  37. Rotate(O_2, ((-((10 * p14) + p18)) * pi / 180), P_2) → T_3
  38. Ray(P_2, T_3) → j_2
  39. Intersect(e_2, j_2, 2) → U_3
  40. Rotate(P_2, ((-((10 * p11) + p17)) * pi / 180), U_3) → V_3
  41. Ray(U_3, V_3) → g_3
  42. Intersect(e_2, g_3, 2) → W_3
  43. Rotate(O_2, (((10 * p15) + p10) * pi / 180), S_2) → Z_3
  44. Ray(S_2, Z_3) → r_3
  45. Intersect(e_2, r_3, 2) → A_4
  46. Point(c_4) → B_4
  47. Point(c_4) → C_4
  48. Point(c_4) → D_4
  49. Segment(B_4, C_4) → h_4
  50. Segment(B_4, D_4) → r_4
  51. Segment(E_4, C_4) → s_4
  52. Segment(E_4, D_4) → t_4
  53. Angle(C_4, E_4, D_4) → ε
  54. Angle(C_4, B_4, D_4) → ζ
  55. Segment(A_2, P_2) → h_2
  56. Segment(O_2, P_2) → k_2
  57. Rotate(P_2, (((100 * p14) + (10 * p18) + p11) * pi / 180), O_2) → G_4
  58. Ray(O_2, G_4) → h_3
  59. Mirror(C_4, E_4) → I_4
  60. Intersect(s_4, t_4) → J_4
  61. Segment(J_4, I_4) → b_4
  62. Angle(D_4, J_4, I_4) → γ
  63. AttachCopyToView(PolyLine[(1.8489,-1.6851), (1.8073,-1.5604), (1.0176,-0.85381), (0.68514,-0.52131), (0.51889,-0.31350), (0.35264,-0.18881), (0.10327,-0.022561), (-0.27079,0.30994), (-0.64485,0.60087), (-0.91501,0.82947), (-1.7670,1.3074), (-2.3074,1.5984), (-2.6398,1.7231), (-2.8892,1.8477), (-3.1594,1.9309), (-3.4711,2.0971), (-3.7205,2.1802), (-4.1777,2.4712), (-4.4270,2.5959), (-4.6348,2.6790), (-4.8842,2.8452), (-5.1336,2.9699), (-5.3414,3.0531), (-5.6323,3.1362), (-5.8817,3.1777), (-6.0895,3.2609), (-6.3597,3.2816), (-6.5467,3.3440), (-6.7130,3.3440), (-6.9416,3.3648), (-7.2533,3.4271), (-7.2533,3.1362), (-7.0455,2.6374), (-6.8792,2.2842), (-7.0455,2.6374), (-7.2117,3.0531), (-7.2741,3.3232), (-7.3780,3.5102), (-7.2325,3.7181), (-7.0455,3.8012), (-6.8792,3.9259), (-6.5883,4.1337), (-6.2973,4.3415), (NaN,NaN), true], 1, (-20.67793434487736, -14.694090934187248), (21.216971492055066, 14.60740570771501), (0, 705), (1008, 0)) → stroke2
  64. AttachCopyToView(PolyLine[(-7.0455,9.6170), (-7.0455,9.4100), (-7.0455,9.1600), (-7.0455,8.9524), (-7.0039,8.7446), (-6.9208,8.5368), (-6.8584,8.3082), (-6.6922,7.9965), (-6.4428,7.6224), (-6.1519,7.3938), (-5.8817,7.2691), (-5.6323,7.2484), (-5.3206,7.3107), (-5.0089,7.5393), (-4.8011,7.6640), (-4.5933,7.8510), (-4.4063,8.1004), (-4.2608,8.3705), (-4.0738,8.5991), (-3.8659,8.6823), (-3.8036,8.4121), (-3.7620,8.1627), (-3.7205,7.9549), (-3.6373,7.7471), (-3.7620,8.0796), (-3.8867,8.4952), (-3.9283,8.7030), (-4.2816,8.6823), (-4.5517,8.6199), (-4.7595,8.6199), (-4.9258,8.6199), (NaN,NaN), true], 1, (-20.67793434487736, -14.694090934187248), (21.216971492055066, 14.60740570771501), (0, 705), (1008, 0)) → stroke1
  65. Rotate(P_2, (((100 * p13) + (10 * p12) + p16) * pi / 180), A_2) → F_4
  66. Segment(A_2, F_4) → f_3
  67. Angle(P_2, A_2, F_4) → α
  68. Rotate(P_2, (((10 * p18) + p11) * pi / 180), O_2) → H_4
  69. Ray(O_2, H_4) → a_4
  70. Intersect(e_2, a_4, 2) → M_4
  71. Angle(P_2, O_2, H_4) → β
  72. Segment(O_2, M_4) → d_4
  73. Rotate(F_4, (((10 * p14) + p17) * pi / 180), A_2) → N_4
  74. Segment(A_2, N_4) → e_4
  75. Angle(F_4, A_2, N_4) → δ

AI Drawing Prompt Reference

Draw a circle with center O and radius r. Select four points A, B, C, and D on the circle and connect them to form quadrilateral ABCD. Choose another point P on the circle and draw line segment OP. Construct angles ∠APB and ∠CPD, and observe their size relationship. Construct an inscribed quadrilateral and demonstrate the relationship between the central angle and the inscribed angle.
The prompt above was compiled from the original drawing commands and is for reference only.