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Equilateral Action (2)! – Special Property of Equilateral Triangles

Equilateral Action (2)! – Special Property of Equilateral Triangles
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2016-07-04 09:21
TriangleCircleAngleTransformationGeometric Construction
① This interactive GeoGebra example demonstrates a special property of equilateral triangles through dynamic construction and exploration. ② Core mathematical concepts include equilateral triangle properties, rotation, translation, dilation (homothety), circle-line intersections, parallel lines, and angle relationships. ③ Construction highlights: using Polygon to create the triangle, Dilate for scaling, Segment and Circle for drawing, Intersect for finding intersection points, Rotate for rotation, Vector and Translate for translation, OrthogonalLine for perpendiculars, Angle for measuring, CircleSector for sectors, Line and Distance for lines and lengths, PointIn for marking points, and a button (button1) to trigger dynamic actions. ④ Suitable for middle and high school geometry classes, this example helps students explore symmetry, transformations, and the use of dynamic geometry software.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 3) → poly1, a, b, c, C
  2. Dilate(A, If[0 < d ≤ 1, 1 - d, 0], B) → A'
  3. Segment(A, A') → e
  4. Dilate(B, If[1 < d ≤ 2, 2 - d, 0], C) → B'
  5. Segment(B, B') → f
  6. Dilate(C, If[2 < d ≤ 3, 3 - d, 0], A) → C'
  7. Segment(C, C') → g
  8. Segment(A, B) → h
  9. Segment(A, C) → i
  10. Segment(B, C) → j
  11. Circle(A, (0.15 * h)) → k
  12. Intersect(k, i, 1) → D
  13. Intersect(k, h, 1) → E
  14. Rotate(A, If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 3), 1.0471975511965976], A) → A'_1
  15. Rotate(A', If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 3), 1.0471975511965976], A) → A''
  16. Segment(A'_1, A'') → e'
  17. Rotate(A, If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * π / 3), -1.0471975511965976], B) → A'_2
  18. Rotate(A', If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * π / 3), -1.0471975511965976], B) → A''_1
  19. Segment(A'_2, A''_1) → e'_1
  20. Dilate(A, If[6 < d ≤ 7, 7 - d, 0], B) → A'_3
  21. Vector(A, A'_3) → u
  22. Circle(B, (0.15 * h)) → r
  23. Intersect(f, e'_1) → G
  24. Intersect(f, e'_1) → H
  25. Intersect(f, e'_1) → I
  26. Intersect(f, e'_1) → J
  27. Intersect(r, j, 1) → F
  28. Intersect(r, h, 1) → K
  29. Dilate(B, If[8 < d ≤ 9, 9 - d, 0], C) → B'_1
  30. Vector(B, B'_1) → v
  31. Intersect(g, e') → L
  32. Polygon(A, B, C) → poly2, c_2, a_1, b_1
  33. PointIn(poly2) → T
  34. OrthogonalLine(T, h) → n
  35. OrthogonalLine(T, i) → g_1
  36. OrthogonalLine(T, j) → h_1
  37. Intersect(g_1, i) → U
  38. Intersect(n, h) → V
  39. Intersect(h_1, j) → W
  40. Angle(T, U, C) → α
  41. Angle(T, V, A) → β
  42. Angle(T, W, B) → γ
  43. Circle(W, (0.1 * h)) → q
  44. Circle(V, (0.1 * h)) → s
  45. Circle(U, (0.1 * h)) → t
  46. Intersect(q, j, 2) → Z
  47. Intersect(q, h_1, 2) → A_1
  48. Intersect(s, h, 2) → B_1
  49. Intersect(s, n, 2) → C_1
  50. Intersect(t, i, 1) → D_1
  51. Intersect(t, g_1, 1) → E_1
  52. CircleSector(W, Z, A_1) → k_1
  53. CircleSector(V, B_1, C_1) → p_1
  54. CircleSector(U, D_1, E_1) → q_1
  55. Dilate(T, If[10 < d ≤ 11, 11 - d, 0], U) → T'
  56. Segment(T, T') → i_1
  57. Dilate(T, If[10 < d ≤ 11, 11 - d, 0], V) → T'_1
  58. Dilate(T, If[10 < d ≤ 11, 11 - d, 0], W) → T'_2
  59. Segment(T, T'_1) → a_2
  60. Segment(T, T'_2) → b_2
  61. Rotate(q_1, If[11.5 < d ≤ 12.5, ((d - 11.5) * π / 2), 1.5708], U) → r_1
  62. Rotate(p_1, If[11.5 < d ≤ 12.5, ((d - 11.5) * π / 2), 1.5708], V) → d_2
  63. Rotate(k_1, If[11.5 < d ≤ 12.5, ((d - 11.5) * π / 2), 1.5708], W) → e_2
  64. Rotate(B, π / 3, A) → B'_2
  65. Dilate(B'_2, If[7 < d ≤ 8, 8 - d, 0], U) → M_1
  66. Segment(B'_2, M_1) → k_2
  67. Dilate(A, If[8 < d ≤ 9, 9 - d, 0], V) → A'_4
  68. Segment(A, A'_4) → l_2
  69. Dilate(B, If[9 < d ≤ 10, 10 - d, 0], W) → B'_3
  70. Segment(B, B'_3) → m_2
  71. Line(A, B) → l
  72. Distance(A, B) → m
  73. Circle(A, (0.15 * m)) → p
  74. OrthogonalLine(A, l) → f_1
  75. Intersect(p, f_1, 1) → M
  76. Line(M, l) → j_1
  77. Circle(A, (0.3 * m)) → c_1
  78. Intersect(c_1, f_1, 1) → N
  79. Line(N, l) → l_1
  80. Circle(A, (0.45 * m)) → d_1
  81. Intersect(d_1, f_1, 1) → O
  82. Line(O, j_1) → m_1
  83. Dilate(A, If[3.5 < d ≤ 4.5, 4.5 - d, 0], M) → A'_5
  84. Vector(A, A'_5) → w
  85. Translate(A, w) → A'_6
  86. Translate(A', w) → A''_2
  87. Segment(A'_6, A''_2) → e'_2
  88. Rotate(A'_6, If[4.5 < d ≤ 5.5, ((d - 4.5) * π), π], A''_2) → R
  89. Rotate(A''_2, If[4.5 < d ≤ 5.5, ((d - 4.5) * π), π], A''_2) → S
  90. Segment(R, S) → n_1
  91. Rotate(R, If[5.5 < d ≤ 6.5, ((d - 5.5) * π), π], R) → R'
  92. Rotate(S, If[5.5 < d ≤ 6.5, ((d - 5.5) * π), π], R) → S'
  93. Segment(R', S') → s_1
  94. Dilate(M, If[12.5 < d ≤ 13.5, 13.5 - d, 0], N) → M'
  95. Vector(M, M') → u_1
  96. Segment(M, S') → t_1
  97. Translate(M, u_1) → M'_1
  98. Translate(S', u_1) → S''
  99. Segment(M'_1, S'') → e_1
  100. Circle(M', e_1 / 4) → f_2
  101. Intersect(f_2, e_1, 1) → F_1
  102. Circle(F_1, e_1 / 4) → g_2
  103. Intersect(g_2, e_1, 2) → G_1
  104. Circle(G_1, e_1 / 4) → h_2
  105. Intersect(h_2, e_1, 2) → H_1
  106. Dilate(M'_1, If[13.5 < d ≤ 14.5, 14.5 - d, 0], F_1) → I_1
  107. Segment(M', F_1) → i_2
  108. Segment(F_1, G_1) → j_2
  109. Segment(G_1, H_1) → n_2
  110. Segment(H_1, S'') → p_2
  111. Segment(M', I_1) → q_2
  112. Rotate(M', If[14.5 < d ≤ 15.5, ((d - 14.5) * π), π], F_1) → M''
  113. Rotate(I_1, If[14.5 < d ≤ 15.5, ((d - 14.5) * π), π], F_1) → J_1
  114. Segment(M'', J_1) → r_2
  115. Rotate(M'', If[15.5 < d ≤ 16.5, ((d - 15.5) * π), π], G_1) → M'''
  116. Rotate(J_1, If[15.5 < d ≤ 16.5, ((d - 15.5) * π), π], G_1) → P
  117. Segment(M''', P) → s_2
  118. Rotate(M''', If[16.5 < d ≤ 17.5, ((d - 16.5) * π), π], H_1) → Q
  119. Rotate(P, If[16.5 < d ≤ 17.5, ((d - 16.5) * π), π], H_1) → P'
  120. Segment(Q, P') → t_2
  121. Dilate(M', If[17.5 < d ≤ 18.5, 18.5 - d, 0], O) → M''_1
  122. Vector(M', M''_1) → v_1
  123. Translate(G_1, v_1) → K_1
  124. Line(M''_1, K_1) → f_3
  125. Circle(M''_1, l_2) → c_3
  126. Intersect(c_3, f_3, 2) → L_1
  127. Circle(L_1, k_2) → d_3
  128. Intersect(d_3, f_3, 2) → N_1
  129. Distance(N_1, K_1) → o
  130. Segment(M''_1, L_1) → g_3
  131. Segment(L_1, N_1) → h_3
  132. Segment(N_1, K_1) → i_3
  133. Dilate(A, If[19 < d ≤ 20, 20 - d, 0], M''_1) → A'_7
  134. Vector(A, A'_7) → w_1
  135. Translate(A, w_1) → A'_8
  136. Translate(A'_4, w_1) → O_1
  137. Segment(A'_8, O_1) → j_3
  138. Dilate(U, If[20 < d ≤ 21, 21 - d, 0], L_1) → U'
  139. Vector(U, U') → u_2
  140. Translate(B'_2, u_2) → P_1
  141. Translate(M_1, u_2) → Q_1
  142. Segment(P_1, Q_1) → k_3
  143. Rotate(P_1, If[21 < d ≤ 22, ((-(d - 21)) * π / 3), ((-π)) / 3], L_1) → R_1
  144. Rotate(Q_1, If[21 < d ≤ 22, ((-(d - 21)) * π / 3), ((-π)) / 3], L_1) → S_1
  145. Segment(R_1, S_1) → l_3
  146. Dilate(B, If[22 < d ≤ 23, 23 - d, 0], N_1) → B'_4
  147. Vector(B, B'_4) → v_2
  148. Translate(B, v_2) → B'_5
  149. Translate(B'_3, v_2) → T_1
  150. Segment(B'_5, T_1) → m_3
  151. Rotate(B'_5, If[23 < d ≤ 24, (((-(d - 23)) * 2) * π / 3), (-2 * π / 3)], N_1) → U_1
  152. Rotate(T_1, If[23 < d ≤ 24, (((-(d - 23)) * 2) * π / 3), (-2 * π / 3)], N_1) → V_1
  153. Segment(U_1, V_1) → n_3

AI Drawing Prompt Reference

Draw an equilateral triangle and demonstrate its rotation, translation, and dilation transformations. Start by constructing the triangle, then rotate one vertex, and obtain new triangles through translation and scaling. Next, draw a circle with one side of the triangle as the diameter, find the intersection points of the circle with the other two sides. Finally, measure the angle relationships between the original and rotated triangles and mark these angles with small circles.
The prompt above was compiled from the original drawing commands and is for reference only.