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Exploring Points of Intersection

Exploring Points of Intersection
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2016-04-03 07:30
FunctionTrigonometryCoordinate GeometrySolid GeometryGeometric Construction
This example demonstrates the construction and analysis of intersections among geometric objects like lines, angle bisectors, circles, and cylinders using GeoGebra. Core mathematical concepts include linear equations, slope differences, angle bisectors, coordinate geometry, and solid geometry. The construction involves commands and objects such as Line, AngularBisector, Intersect, OrthogonalLine, Circle, Segment, Mirror, Dilate, Midpoint, and Center as outlined in the construction summary. It is suitable for high school students and educators for deepening understanding of geometric transformations and equation solving.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. AngularBisector(C, A, B) → d
  5. AngularBisector(A, B, C) → e
  6. AngularBisector(B, C, A) → f
  7. Intersect(d, e) → D
  8. AngularBisector(C, A, D) → g
  9. AngularBisector(D, C, A) → h
  10. Intersect(g, h) → E
  11. OrthogonalLine(E, c) → i
  12. Intersect(c, i) → F
  13. Circle(E, F) → k
  14. AngularBisector(D, A, B) → j
  15. AngularBisector(D, B, A) → l
  16. Intersect(j, l) → G
  17. OrthogonalLine(G, a) → m
  18. Intersect(a, m) → H
  19. Circle(G, H) → p
  20. AngularBisector(B, C, D) → n
  21. AngularBisector(D, B, C) → q
  22. Intersect(n, q) → I
  23. OrthogonalLine(I, b) → r
  24. Intersect(b, r) → J
  25. Circle(I, J) → s
  26. Intersect(k, d) → K, N
  27. Intersect(p, d) → L, M
  28. Segment(E, G) → t
  29. Mirror(d, t) → d'
  30. Segment(E, I) → a_1
  31. Segment(I, G) → b_1
  32. Mirror(e, b_1) → e'
  33. Mirror(f, a_1) → f'
  34. Intersect(d', e') → O
  35. AngularBisector(O, F, A) → c_1
  36. AngularBisector(A, H, O) → d_1
  37. Intersect(d, c_1) → P
  38. OrthogonalLine(P, a) → e_1
  39. Intersect(a, e_1) → Q
  40. Circle(P, Q) → f_1
  41. AngularBisector(O, J, B) → g_1
  42. Intersect(e, g_1) → R
  43. OrthogonalLine(R, a) → h_1
  44. Intersect(a, h_1) → S
  45. Circle(R, S) → k_1
  46. AngularBisector(O, J, C) → i_1
  47. Intersect(i_1, f) → T
  48. OrthogonalLine(T, b) → j_1
  49. Intersect(b, j_1) → U
  50. Circle(T, U) → p_1
  51. Intersect(p_1, k_1) → V, , ,
  52. OrthogonalLine(V, b) → l_1
  53. Intersect(b, l_1) → W
  54. Circle(V, W) → q_1
  55. Dilate(q_1, 0.5, W) → r_1
  56. Intersect(k_1, q_1) → Z, A_1, ,
  57. Line(A_1, Z) → m_1
  58. Midpoint(V, W) → B_1
  59. Line(B_1, m_1) → n_1
  60. Intersect(r_1, n_1) → C_1, D_1
  61. Mirror(r_1, D_1) → s_1
  62. Mirror(s_1, q_1) → t_1
  63. Intersect(f_1, p_1, 1) → E_1
  64. OrthogonalLine(E_1, c) → a_2
  65. Intersect(c, a_2) → F_1
  66. Midpoint(F_1, E_1) → G_1
  67. Circle(G_1, F_1) → c_2
  68. Circle(E_1, F_1) → d_2
  69. Intersect(f_1, d_2) → H_1, I_1, ,
  70. Line(I_1, H_1) → b_2
  71. Line(G_1, b_2) → e_2
  72. Intersect(c_2, e_2) → J_1, K_1
  73. Mirror(c_2, J_1) → f_2
  74. Mirror(f_2, d_2) → g_2
  75. Intersect(f_1, k_1, 1) → L_1
  76. OrthogonalLine(L_1, a) → h_2
  77. Intersect(a, h_2) → M_1
  78. Midpoint(M_1, L_1) → N_1
  79. Circle(L_1, M_1) → k_2
  80. Circle(N_1, M_1) → p_2
  81. Intersect(k_2, f_1) → O_1, P_1, ,
  82. Line(P_1, O_1) → i_2
  83. Line(N_1, i_2) → j_2
  84. Intersect(p_2, j_2) → Q_1, R_1
  85. Mirror(p_2, Q_1) → q_2
  86. Mirror(q_2, k_2) → r_2
  87. Center(f_1) → S_1
  88. Center(p_1) → T_1
  89. Center(k_1) → U_1
  90. Center(r_2) → V_1
  91. Center(g_2) → W_1
  92. Center(t_1) → Z_1
  93. Segment(Z_1, A) → l_2
  94. Segment(C, V_1) → m_2
  95. Intersect(l_2, m_2) → A_2
  96. Segment(W_1, B) → n_2