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Dynamic Geometry: Equilateral Triangles and Pyramids

Dynamic Geometry: Equilateral Triangles and Pyramids
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2018-02-09 00:57
TriangleCircleVectorGeometric Construction
This example dynamically illustrates two problems from different sources (Antonio Gutierrez and Alexander Bogomolny) involving equilateral triangles and pyramids. It covers core concepts like properties of equilateral triangles, unit circle, midpoints, vectors, and transformations. Construction highlights include Rotate, Dilate, Polygon, Midpoint, and Vector commands in GeoGebra. Suitable for high school and above learners exploring geometric transformations and spatial relationships.

GeoGebra Commands & Parameters

  1. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  2. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  3. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  4. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  5. Midpoint(A, C) → V
  6. CircleArc(V, C, D) → g_2
  7. Point(g_2) → B
  8. Segment(C, B) → j_1
  9. Segment(A, B) → n
  10. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  11. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  12. Segment(R, A''') → f''
  13. Intersect(n, f'') → Z
  14. Segment(B, Z) → l_1
  15. Segment(A, A') → f
  16. Segment(A'_1, A'') → f'
  17. Segment(A, B_2) → g
  18. Segment(B_2, C) → h
  19. Segment(C, D) → i
  20. Segment(D, A) → j
  21. Circle(B_2, (0.15 * f)) → p
  22. Circle(C, Radius[p]) → k
  23. Circle(A, (0.1 * j)) → q
  24. Circle(D, (0.1 * j)) → r
  25. Segment(A, C) → l
  26. Segment(D, B_2) → m
  27. Intersect(r, j, 1) → F
  28. Intersect(r, m, 1) → G
  29. Intersect(r, i, 1) → H
  30. Intersect(k, i, 1) → I
  31. Intersect(k, l, 1) → J
  32. Intersect(k, h, 1) → K
  33. Intersect(p, h, 1) → L
  34. Intersect(p, m, 1) → M
  35. Intersect(p, g, 1) → N
  36. Intersect(q, g, 1) → O
  37. Intersect(q, l, 1) → P
  38. Intersect(q, j, 1) → Q
  39. CircleSector(B_2, L, N) → s
  40. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  41. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  42. Segment(R', S) → f'''
  43. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  44. Vector(B_2, B') → u
  45. Translate(s, u) → s'
  46. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  47. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  48. Vector(C, C') → v
  49. Translate(s'', v) → s'''
  50. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  51. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  52. Vector(D, D') → w
  53. Translate(t, w) → t'
  54. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  55. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  56. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  57. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  58. Vector(B_2, B'_1) → v_1
  59. Vector(B_2, B'_3) → w_1
  60. Vector(B_2, B'_2) → i_1
  61. Circle(A, n - j_1) → h_2
  62. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  63. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  64. Vector(B_2, B'_4) → u_1
  65. Translate(s, u_1) → s'_1
  66. Line(B, f') → a_1
  67. Point(a_1) → U
  68. Angle(U, B, C) → α
  69. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  70. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  71. Line(A, B_2) → h_1
  72. Angle(B, A, D) → β
  73. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  74. Mirror(C, D) → C'_1
  75. Rotate(C, ((-π)) / 3, B_2) → C'_2
  76. Segment(B_2, C'_2) → f_1
  77. Segment(C, C'_2) → g_1
  78. Intersect(k, g_1, 1) → E
  79. Intersect(k, h) → T,
  80. CircleSector(C, T, E) → d_1
  81. Rotate(D, 15°, C) → D'_2
  82. Angle(D, C, D'_2) → γ
  83. Ray(C, D'_2) → k_1
  84. Rotate(C, (-(15°)), D) → C'_3
  85. Angle(C'_3, D, C) → δ
  86. Ray(D, C'_3) → m_1
  87. Intersect(k_1, m_1) → W
  88. Rotate(B_2, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → A_1
  89. Rotate(C, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → C'_4
  90. Segment(A_1, C'_4) → h'
  91. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  92. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  93. Segment(B_1, C'_5) → h'_1
  94. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  95. CircleSector(C, E', E) → c_1
  96. Polygon(C, W, D) → poly2, d_2, c_2, w_2
  97. Polygon(A, W, D) → poly3, d_3, a_2, w_3
  98. Polygon(B_2, W, C) → poly4, c_3, b_2, w_4
  99. Polygon(A, B_2, W) → poly5, w_5, a_3, b_1
  100. Polygon(B_2, C'_2, C) → poly6, c_4, b_3, c'_2
  101. Midpoint(h') → C_1
  102. Segment(C, C_1) → n_1
  103. Intersect(k, n_1) → D_1,
  104. Dilate(C, If[12 < e ≤ 13, 13 - e, 0], C_1) → C'_6
  105. Segment(C, C'_6) → p_1
  106. Rotate(T, 15°, C) → T'
  107. Angle(T, C, T') → ε
  108. Ray(C, T') → q_1
  109. Intersect(h', q_1) → E_1
  110. Rotate(c_1, If[13.5 < e ≤ 14.5, (((-π)) / 6 * (e - 13.5)), ((-π)) / 6], C) → e_1
  111. Dilate(C, If[16 < e ≤ 17, 17 - e, 0], E_1) → C'_7
  112. Segment(C, C'_7) → r_1
  113. Rotate(D_1, If[16 < e ≤ 17, (((-π)) / 12 * (e - 16)), ((-π)) / 12], C) → F_1
  114. CircleSector(C, F_1, D_1) → s_1
  115. Rotate(s_1, If[17.5 < e ≤ 18.5, (((-π)) / 12 * (e - 17.5)), ((-π)) / 12], C) → t_1
  116. Rotate(t_1, If[19.5 < e ≤ 20.5, (((-π)) / 2 * (e - 19.5)), ((-π)) / 2], C) → e_2
  117. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  118. Vector(C, C'_8) → u_2
  119. Translate(e_2, u_2) → k_2
  120. Rotate(k_2, If[22 < e ≤ 23, ((π - π / 12) * (e - 22)), π - π / 12], D) → p_2
  121. Dilate(D, If[23 < e ≤ 24, 24 - e, 0], W) → D'_3
  122. Dilate(C, If[23 < e ≤ 24, 24 - e, 0], W) → C'_9
  123. Segment(C, C'_9) → f_2
  124. Segment(D, D'_3) → i_2
  125. Dilate(B_2, If[25 < e ≤ 26, 26 - e, 0], W) → G_1
  126. Dilate(A, If[25 < e ≤ 26, 26 - e, 0], W) → A'_2
  127. Segment(A, A'_2) → j_2
  128. Segment(B_2, G_1) → l_2
  129. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  130. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  131. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  132. Polygon(H_1, I_1, C'_{10}) → poly6', c'_{10}, h_3, i_3
  133. Rotate(C, ((-π)) / 3, B_2) → C'_{11}
  134. Dilate(D, If[30 < e ≤ 31, 31 - e, 0], W) → D'_4
  135. Segment(DD, D'_4) → m_2
  136. Dilate(W, If[31 < e ≤ 32, 32 - e, 0], C'_{11}) → W'_1
  137. Segment(W, W'_1) → n_2