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Dynamic Illustration of Square, Equilateral Triangle, and Circumcircle Relations

Dynamic Illustration of Square, Equilateral Triangle, and Circumcircle Relations
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2017-05-18 03:06
CirclePolygonTrigonometryCalculusGeometric Construction
This example dynamically illustrates the geometric relationship among a square, an equilateral triangle, and the circumcircle of the triangle. Key math concepts include circumcircle, equilateral triangle, square, tangent line, reflection, calculus, and differential equations. The construction utilizes GeoGebra commands such as Rotate, Dilate, Polygon, Midpoint, CircleArc, Point, Segment, Intersect, Circle, CircleSector, Vector, Translate, Line, Angle, Mirror, Ray, and Center. It is suitable for high school and above students studying geometric transformations, coordinate geometry, and calculus applications.

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. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  91. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  92. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  93. Polygon(A, B_2, W) → poly5, w_5, a_3, b_1
  94. Rotate(T, 15°, C) → T'
  95. Angle(T, C, T') → ε
  96. Ray(C, T') → q_1
  97. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  98. Vector(C, C'_8) → u_2
  99. Dilate(D, If[23 < e ≤ 24, 24 - e, 0], W) → D'_3
  100. Dilate(C, If[23 < e ≤ 24, 24 - e, 0], W) → C'_9
  101. Dilate(B_2, If[25 < e ≤ 26, 26 - e, 0], W) → G_1
  102. Dilate(A, If[25 < e ≤ 26, 26 - e, 0], W) → A'_2
  103. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  104. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  105. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  106. Rotate(B_2, π / 3, A) → C_1
  107. Polygon(A, C_1, B_2, C, D) → poly2, a_2, c_1, b_2, c_2, d_2
  108. Rotate(A, If[10 < e ≤ 11, ((-(e - 10)) * π / 3), ((-π)) / 3], B_2) → A'_3
  109. Rotate(A', If[10 < e ≤ 11, ((-(e - 10)) * π / 3), ((-π)) / 3], B_2) → A''_1
  110. Segment(A'_3, A''_1) → f'_1
  111. Rotate(A, If[11 < e ≤ 12, ((e - 11) * π / 3), π / 3], A) → A'_4
  112. Rotate(A', If[11 < e ≤ 12, ((e - 11) * π / 3), π / 3], A) → A''_2
  113. Segment(A'_4, A''_2) → f'_2
  114. Polygon(A, B_2, A'_3) → poly3, a'_3, a_4, b_3
  115. Circle(A, A'_3, B_2) → e_1
  116. Rotate(B_2, If[12 < e ≤ 14, ((e - 12) * π), (2 * π)], Center[e_1]) → D_1
  117. Center(e_1) → E_1
  118. CircleArc(E_1, B_2, D_1) → p_1
  119. Midpoint(C, B_2) → F_1
  120. Segment(B_2, F_1) → r_1
  121. Segment(B_2, C) → s_1
  122. Intersect(e_1, s_1) → K_1, L_1
  123. Angle(B_2, E_1, L_1) → ζ
  124. Dilate(C, If[14 < e ≤ 15, 15 - e, 0], L_1) → C'_6
  125. Segment(C, C'_6) → t_1
  126. Rotate(C, If[15 < e ≤ 16, ((-(e - 15)) * π / 3), ((-π)) / 3], L_1) → C'_7
  127. Rotate(C'_6, If[15 < e ≤ 16, ((-(e - 15)) * π / 3), ((-π)) / 3], L_1) → J_1
  128. Segment(C'_7, J_1) → k_2
  129. Dilate(L_1, If[16 < e ≤ 17, 17 - e, 0], B_2) → M_1
  130. Vector(L_1, M_1) → v_2
  131. Translate(C'_7, v_2) → N_1
  132. Translate(J_1, v_2) → O_1
  133. Segment(N_1, O_1) → m_2
  134. Rotate(N_1, If[17 < e ≤ 18, ((e - 17) * π / 3), π / 3], B_2) → P_1
  135. Rotate(O_1, If[17 < e ≤ 18, ((e - 17) * π / 3), π / 3], B_2) → Q_1
  136. Segment(P_1, Q_1) → n_2
  137. Line(D, P_1) → n_1
  138. Intersect(f'_2, n_1) → R_1
  139. Circle(R_1, A'_3) → e_2
  140. Circle(R_1, (0.7 * Radius[e_2])) → p_2
  141. Intersect(p_2, n_1) → S_1, T_1
  142. Intersect(p_2, f'_2) → U_1, V_1
  143. Dilate(D, If[18 < e ≤ 19, 19 - e, 0], R_1) → D'_4
  144. Segment(D, D'_4) → q_2
  145. Dilate(R_1, If[19 < e ≤ 20, 20 - e, 0], P_1) → W_1
  146. Segment(R_1, W_1) → r_2
  147. Rotate(S_1, If[20 < e ≤ 21, ((e - 20) * π / 2), π / 2], R_1) → Z_1
  148. CircleSector(R_1, S_1, Z_1) → s_2
  149. Rotate(s_2, If[21.75 < e ≤ 22.75, ((e - 21.75) * π / 2), π / 2], R_1) → t_2