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Concurrent Relationship in Trapezoid and Its 2 Exircles

Concurrent Relationship in Trapezoid and Its 2 Exircles
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2017-05-09 21:14
CirclePolygonGeometric Construction
This example illustrates the concurrent relationship between a trapezoid and its two excircles. By dynamically adjusting, one can observe how the concurrency changes under different conditions. Key mathematical concepts include geometric construction, circle properties, and concurrency. The construction uses commands like Line, Ray, Point, Segment, Circle, Intersect, etc. Suitable for high school geometry students and teachers.

GeoGebra Commands & Parameters

  1. Line(A, B) → f
  2. Ray(A, B) → g
  3. Point(g) → C
  4. Ray(A, C) → h
  5. Segment(A, C) → i
  6. Slope(f) → m
  7. PointIn(a) → D
  8. Segment(A, D) → j
  9. Line(D, i) → k
  10. Point(k) → E
  11. Ray(D, E) → l
  12. Point(l) → F
  13. Ray(D, F) → n
  14. Segment(F, C) → p
  15. Segment(D, F) → q
  16. Min(j, i) → c
  17. Circle(A, (0.2 * c)) → d
  18. Intersect(d, j, 1) → G
  19. Intersect(d, i, 1) → H
  20. CircleSector(A, H, G) → e
  21. Angle(C, A, D) → α
  22. CircleSector(A, G, H) → r
  23. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], C) → A'
  24. Segment(A, A') → s
  25. Dilate(F, If[0 < o ≤ 1, 1 - o, 0], D) → F'
  26. Segment(F, F') → t
  27. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], D) → A'_1
  28. Segment(A, A'_1) → f_1
  29. Ray(A, D) → g_1
  30. Point(g_1) → K
  31. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], D) → A'_2
  32. Vector(A, A'_2) → u
  33. Translate(e, u) → e'
  34. Polygon(A, C, F, D) → poly1, a_1, c_1, f_2, d_1
  35. Translate(r, u) → r'
  36. Dilate(C, If[1 < o ≤ 2, 2 - o, 0], F) → C'_3
  37. Segment(C, C'_3) → i_1
  38. Line(C, F) → j_1
  39. Line(A, D) → k_1
  40. Segment(A, F) → h_1
  41. Segment(D, C) → l_1
  42. Intersect(h_1, l_1) → I
  43. Point(k_1) → J
  44. Angle(F, D, J) → β
  45. Point(j_1) → L
  46. Angle(L, F, D) → γ
  47. Rotate(J, (-(β / 2)), D) → J'
  48. Angle(J', D, J) → ε
  49. Ray(D, J') → m_1
  50. Rotate(L, γ / 2, F) → L'
  51. Angle(L, F, L') → δ
  52. Ray(F, L') → n_1
  53. Intersect(m_1, n_1) → M
  54. OrthogonalLine(M, j_1) → p_1
  55. OrthogonalLine(M, t) → q_1
  56. OrthogonalLine(M, k_1) → r_1
  57. Intersect(j_1, p_1) → N
  58. Intersect(q_1, t) → O
  59. Intersect(r_1, k_1) → P
  60. Circle(M, N) → e_1
  61. Point(k_1) → Q
  62. Point(j_1) → R
  63. Angle(Q, A, C) → ζ
  64. Angle(A, C, R) → κ_1
  65. Rotate(Q, ζ / 2, A) → Q'
  66. Angle(Q, A, Q') → θ
  67. Rotate(R, (-(κ_1 / 2)), C) → R'
  68. Angle(R', C, R) → η
  69. Ray(C, R') → s_1
  70. Ray(A, Q') → t_1
  71. Intersect(s_1, t_1) → S
  72. OrthogonalLine(S, j_1) → b_1
  73. OrthogonalLine(S, k_1) → g_2
  74. OrthogonalLine(S, s) → h_2
  75. Intersect(j_1, b_1) → T
  76. Intersect(h_2, s) → U
  77. Intersect(g_2, k_1) → V
  78. Circle(S, T) → c_2
  79. Segment(O, U) → i_2
  80. Intersect(j_1, k_1) → W
  81. Dilate(D, If[3 < o ≤ 4, 4 - o, 0], M) → D'
  82. Dilate(F, If[3 < o ≤ 4, 4 - o, 0], M) → F'_1
  83. Segment(D, D') → j_2
  84. Segment(F, F'_1) → k_2
  85. Circle(D, Radius[d]) → d_2
  86. Circle(F, Radius[d]) → e_2
  87. Intersect(e_2, n_1) → Z,
  88. Intersect(d_2, m_1) → B_1,
  89. Intersect(d_2, t) → C_1,
  90. CircleSector(D, C_1, B_1) → q_2
  91. Intersect(e_2, j_1) → D_1, E_1
  92. CircleSector(F, E_1, Z) → p_2
  93. Rotate(p_2, If[4.5 < o ≤ 5.5, ((o - 4.5) * δ), δ], F) → r_2
  94. Rotate(q_2, If[5.5 < o ≤ 6.5, ((o - 5.5) * ε), ε], D) → s_2
  95. Circle(C, Radius[d]) → t_2
  96. Intersect(d, t_1) → F_1,
  97. Intersect(d, s) → G_1,
  98. Intersect(t_2, s_1, 1) → A_1
  99. Intersect(t_2, s) → H_1,
  100. CircleSector(A, F_1, G_1) → c_3
  101. CircleSector(C, H_1, A_1) → d_3
  102. Angle(F_1, A, G_1) → κ
  103. Angle(H_1, C, A_1) → λ
  104. Dilate(A, If[6.5 < o ≤ 7.5, 7.5 - o, 0], S) → A'_3
  105. Dilate(C, If[6.5 < o ≤ 7.5, 7.5 - o, 0], S) → C'
  106. Segment(A, A'_3) → l_2
  107. Segment(C, C') → m_2
  108. Rotate(c_3, If[8 < o ≤ 9, ((-(o - 8)) * κ), (-κ)], A) → e_3
  109. Rotate(d_3, If[9 < o ≤ 10, ((o - 9) * λ), λ], C) → f_3
  110. Dilate(S, If[10 < o ≤ 11, 11 - o, 0], U) → S'
  111. Dilate(M, If[10 < o ≤ 11, 11 - o, 0], O) → M'
  112. Segment(S, S') → n_2
  113. Segment(M, M') → a_2
  114. Angle(M, O, D) → ι
  115. Angle(A, U, S) → μ
  116. Rotate(O, If[11 < o ≤ 13, ((o - 11) * π), (2 * π)], M) → O'
  117. Rotate(U, If[11 < o ≤ 13, ((o - 11) * π), (2 * π)], S) → U'
  118. CircleArc(M, O, O') → g_3
  119. CircleArc(S, U, U') → h_3
  120. Dilate(A, If[13 < o ≤ 14, 14 - o, 0], F) → A'_4
  121. Segment(A, A'_4) → b_2
  122. Dilate(C, If[14 < o ≤ 15, 15 - o, 0], I) → C'_1
  123. Segment(C, C'_1) → i_3
  124. Dilate(I, If[15 < o ≤ 16, 16 - o, 0], D) → I'
  125. Segment(I, I') → j_3
  126. Dilate(U, If[16.5 < o ≤ 17.5, 17.5 - o, 0], I) → U'_1
  127. Segment(U, U'_1) → k_3
  128. Dilate(I, If[17.5 < o ≤ 18.5, 18.5 - o, 0], O) → I'_1
  129. Segment(I, I'_1) → l_3